it'^/i)pgBjMga^i;ii,i?;■W^^■.'f^
3tt;ara. ^tm $ark
...Xohn Henr-^.Tanner...
Date Due
5
AUl 1 958
FlB 1 5 '
%Q-
-mf.
tiEe^
H»
^sm
■i^
tff»
&
Cornell University Library
QA 371.J71 1890
A treatise on ordinary and partial diffe
3 1924 001 126 204
Cornell University
Library
The original of tiiis book is in
tine Cornell University Library.
There are no known copyright restrictions in
the United States on the use of the text.
http://www.archive.org/details/cu31924001126204
X
A TREATISE '
as the equation of which equation (i) is the primitive.
Again, equation (2), from which a has disappeared by differ-
entiation, is itself the equation derived from equation (i) as a
primitive, when a is regarded as an arbitrary, and t as a fixed,
constant. But, if both a and c are arbitrary, differentiating
again, we have
^ = 0;
dx'
and, c having disappeared, this is the equation of the second
order of which equation (i) is the primitive.
It is evident that, in every case, the number of differentia-
tions necessary, and therefore the index of the order of the
differential equation produced, will be the same as the number
of constants to be eliminated.
§ I.] NUMBER OF ARBITRARY CONSTANTS. 7
8. Considering now the differential equation as given, the
primitive is an integral equation which satisfies it, the constants
eliminated being, in the reverse process of finding the integral,
the constants of integration ; and it is the most general solution,
or complete integral, because no greater number of constants
could be eliminated without introducing derivatives higher than
the highest which occurs in the given equation. For example,
the process given in the preceding article shows that
c^x — try + a = o
is the complete integral of
x( -^) — y-^ 4- « = o.
\axl ax
It was shown in Art. 4 that this differential equation is
satisfied by j/^ =: /^x, which, it will be noticed, is not a particu-
lar case of the complete integral. Thus, while the complete
integral is the most general solution, it does not, in all cases,
include all the solutions.
9. We thus see that the complete integral of a differential
equation of the first order should contain one constant of
integration, that of an equation of the second order should
contain two constants, and so on. It is, of course, to be under-
stood that no two of the constants admit of being replaced by
a single one. For example, the constants C and a in the equa-
tion
y = Cf^ + "
are equivalent to a single arbitrary constant ; for, putting
A = Ce°-, the equation may be written
y = Ae^,
hence it is the complete integral of an equation of the first,
not of the second, order.
DIFFERENTIAL EQUATIONS. [Art. 10,
Geometrical Illustration of the Meaning of a Differential Equation.
10. Let X and ;;/ in a differential equation be regarded as
the rectangular coordinates of a point in a plane ; then the
derivative — is the tangent of the inclination to the axis of x
dx
of the direction in which the point (x, y) is moving. Putting
i,- ^
^~ dx'
a differential equation of the first order is a relation between the
variables x, y, and /, of which x and y determine the position
of the point, and p the direction of its motion. We may assign
to X and y any values we choose, and then determine from the
equation one or more values of /. We cannot, therefore, regard
the differential equation as satisfied by certain points (that is,
by certain associated values of x and y) ; but it is satisfied by
certain associated values of x, y, and /, that is, by a point in
any position, provided it is moving in the proper direction.
11. Let us now suppose the point (x, y) to start from any
assumed initial position, and to move in the proper direction.
We have thus a moving point satisfying the differential equa-
tion. As the point moves, the values of x and y vary, so that
the value of / derived from the equation will likewise, in gen-
eral, vary ; and we may suppose the direction of the point's
motion to vary in such a way that the moving point contin' es
to satisfy the differential equation. The line which the point
now describes is, in general, a curve ; and the point may evi-
dently move along this curve in either direction, and yet always
satisfy the differential equation. The moving point may return
to its initial position, thus describing a closed curve ; or it may
pass to infinity in both directions, describing an infinite branch
of a curve.
§ I.] GEOMETRICAL REPRESENTATION. 9
If, now, we can determine the equation of this curve in the
form of an ordinary equation between x and y, the value of -^
found by differentiating the equation of the curve will, by
hypothesis, be identical, for any values of x and y, with the
value of p corresponding to the same values of x and y in
the differential equation. The equation of the curve will, there-
fore, be a solution, or integral, of the differential equation.
12. But, since this integral equation restricts the point to
certain positions, the assemblage of which constitutes the
curve, it is not the complete solution of the differential equa-
tion ; for the complete solution ought to represent the moving
point satisfying the equation in all its possible positions. If,
now, we take for initial point any point not on the curve already
determined, and proceed in like manner, we shall determine
another curve, whose equation will be another particular solu-
tion, or integral, of the differential equation. We thus have an
unlimited number of curves forming a system, of curves, and the.
complete integral is the general equation of this system.
This general equation must contain, besides x and y, a
quantity independent of x and y called the parameter, by giving
different values to which we obtain the equations of all the
particular curves of the system. The arbitrary parameter of
the system is, of course, the constant of integration.
13. We may illustrate this by a simple example. Let the
differential equation be
^= -^ (i)
dx y
.Since - is the tangent of the inclination to the axis of x of the
line joining the point {x, y) to the origin, the equation expresses
that the point {x, y) is always moving in a direction perpendicu-
lar to the line joining it to the origin. Starting from any initial
lO DIFFERENTIAL EQUATIONS. [Art. 1 3.
position, it is clear that the point describes a circle about the
origin as centre. The system of curyes in this case, therefore,
consists of all circles whose centres are at the origin ; and the
general equation of this system,
x' -\. y-^ = C, (2)
where C is the parameter, is the complete integral.
Now consider the moving point when in any special posi-
tion, as, for instance, in the position (3, 2) ; we find, by substi-
tuting these values for x and y in equation (2),
C = 13.
Hence
X-' -\- y^ = 13
is the equation of the particular curve in which the point is
then moving. If we differentiate this equation, we find a value
■for -^ at the point (3, 2) identical with that given for the same
dx
point by equation (i).
Doubly Infinite Systems of Curves.
14. In the case of a differential equation of the second order,
let
~- = p and -^- = a;
dx ^ dx" ^ '
then the equation is a relation between x, y, p, and q. It is
possible to assign any values we please to x, y, and p, and to
determine from the equation a value of q, which, in connection
with the assumed values of x, y, and /, will satisfy the equation.
This value of q, in connection with the assumed value of p,
determines the curvature of the path of the moving point
§ I] GEOMETRICAL REPRESENTATION. II
{x, y). Hence a differential equation of the second order may-
be regarded as satisfied by a moving point having any assumed
position, and moving in any assumed direction, provided only
that its path have the proper curvature. Starting from any
assumed initial point, and in any assumed initial direction, the
point {x, y) may move in such a manner as to satisfy the equa-
tion. As it moves the values of x and y will vary ; and, since
the path has a definite curvature at this point, the value of /
will likewise vary. Hence the value of q derived from the
differential equation will, in general, also vary ; but we may
suppose the curvature of the path to vary in such a manner
that the moving point continues to satisfy the equation. A
curve is thus described whose ordinary equation is a solution
of the differential equation, since the simultaneous values of x,
y, -^, and —^, at every point of it, by hypothesis, satisfy that
dx dx^
equation.
15. As before, the complete integral is the general equation
of the system of curves which may be generated in the manner
explained above ; but this system has a greater generality than
that which represents a differential equation of the first order.
For, in its general equation, it must be possible to assign any
assumed simultaneous values to x, y, and p. Substituting the
assumed values in the general equation and in the result of its
differentiation, we have two equations ; and, in order to satisfy
them, we must have two arbitrary parameters at our disposal. -
The system of curves representing a differential equation of
the second order is, therefore, a system containing two param-
eters, to each of which independently an unlimited number of
values may be assigned. Such a system is said to be a doubly
infinite system of curves.
In like manner, it may be shown that a differential equation
of the third order is represented by a triply infinite system of
curves, and so on.
12 DIFFERENTIAL EQUATIONS. [Art. 1 5.
Examples I.
1. Form the differential equation of which y = ^ cos x is the com-
plete integral.
-^ + ^tan^ = o.
dx
2. Form the equation of which y = ax' + ix is the complete inte-
gral, a and d being arbitrary.
, d'y dy ,
or — '^ — 2x^- + 2y = o.
dx' dx
3. Form the equation of which y — 2cx — <:^ = o is the complete
integral.
y\-f\ -f 2^-/ — J = o.
\dxl dx
4. Form the equation of which e'^y -\- 2cxey -\-c'= o is the
primitive.
(|)-(. -«.)+. = o.
5. Form the equation of which y = {x ■\- (r)f«^ is the complete
integral.
-^ = e"^ -\- ay.
dx
6. Denoting by 6 the inclination to the axis of x of the line joining
{x, y') to the origin, and by the inchnation of the point's motion, write
the differential equation which expresses that is the supplement of Q
and show that it represents a system of hyperbolas.
7. With the same notation, write the differential equation which
expresses that cf) = 2O, and show that it represents a system of circles
passing through the origin.
§ I.] EXAMPLES. . 13
8^ Form the differential equation of the system of straight lines
which touch the circle oc^ -\- f- = i, and show that this circle also
satisfies the equation.
9. Find the differential equation of all the circles having their radii
equal to a.
10. Find the differential equation of all the conies whose axes
coincide with the coordinate axes.
dy I dy\- d^y
ax \dxj ar"
14 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. l6.
CHAPTER II.
EQUATIONS OF THE FIRST ORDER AND DEGREE,
II.
Separation of the Variables.
i6. In an equation of the first order, it is immaterial whether
;ir or _y be taken as the independent variable. If the equation
is also of the first degree, it is frequently written in the form
Mdx + Ndy = o,
in which M and N denote functions of x and y. The simplest
case is that in which the equation may be so written that the
coefficient oi dx isa. function of x only, and that oi j a. function
of J only ; in other words, the case in which the equation can
be written in the form
f{x)dx + (f>{y)dy = o (i)
The complete integral is then evidently
yix)dx+^(l>{y)dy = C. (2)
§11.] SEPARATION OF THE VARIABLES. IS
17. The process of reducing an equation, when possible, to
the form (i) is called the separation of the variables. For
example, in the equation
(i — y)dx + (i + x)dy = 0, (i)
the variables are separated by dividing by (i — y){\ + x) ;
thus,
^■^ + ^y ^o (2)
Hence, integrating,
log(i + ;«)- log(i - jc) = ' ^ '
for, substitute in this
" = ^tM {^)
y = r] + k,)
and we have
dr] _ a^ + irj + ah + M + c ^ s
di ~ a'i + l>'r, + a'h + i>'k + c' ^^^
If, now, we determine A and k by the equations
ah + Ilk -{- c = o,
'=:;! <^>
a'h + d'k + c'
equation (2) takes the homogeneous form
dr] at -\- brt
dk~ a'i + b'-r{
from which we can determine the integral relation between ^
and 17; and thence, by substitution from (2), the relation
between x and y.
24. Equations (4) give impossible values of h and k when a,
b, d, and b' are proportional. In this case, putting
d = ma, b' = mh.
20 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 24.
i
equation (i) becomes
dy _ ax -Y by ■\- c
dx m{ax + by^ + d'
Now put
^x ■\- by = z;
whence
dy _ \ dz a
dx b dx b'
Making these substitutions, we have
djL^a^b^±-^,
dx mz + (f
an equation in which the variables can be separated.
Examples II.
Solve the following differential equations : —
I. (i + x)ydx + (i — y)xdy = 0, \o%xy = c — x -\- y.
dy _ 2
'dx~ "'^ ^' ^■*^'-^ + (y + 2 = o.
3. {y^ + xy-)dx + {x^ ~ yx^)dy = o, W- _ -^ + ^ ^ ^
° y xy
4. xy{i + x^)dy = (i + y')dx, (i + ^)(i + y^) = cx\
xdx ydy
dx by — a
dy f + -i
'^' dx^ x^ + 1' y - X = c(i + xy).
II.] EXAMPLES. 21
8. sin X cos y . dx ■= cos * sin ^ . dy, cos y ^= c cos x.
9. fla:-=^ 4- 2J' = ^jc— , x'y'^ = ff^".
10. i^ + ^ + y + y" _ o a; + jC + I _ ^
^jc I + ^ + x' 2ar)' + X + y — I
11. --^ + ^j;)/ = ^^j);% log- ^ ~ ^ = ^^ + c.
12. a;^(i — y) = y(i + x"), log- -i- = c.
dx X a
13. ar(^ — ydx — sJlyX^ + y^)dx = 0, a:^ = ^ + 2fji'.
14. {2>y + iox)dx + (sj; + ']x)dy = o.
(j' + xYiy + 2;e)3 = c.
15. (:«: + jv)-^' -{- X — y = o, tan-'^ + ^log(*^ + f) = c.
dx X
16. {xy - ^)£ = jv% y^ J.
17. .» + J-/ = 2y, log (^ - j(;) = c ^— .
18. (3JI' — 7«'+ 7)^^ + (7J>' — S'* + 3)'^ = Oj
(y — X + ^y(y + jc — 1)5 = (T.
19. (x^ + y)dx — 2:cy(^ = 0, ^2 — j);2 = ex.
20. 2;ri'a'x + (j^ — 5x^)dy = 0, jc' — j^ = ^rys.
21. y + (xy + x^)^ = o, a;;'^ = ^{x + 2y).
dx
22. (x' — 5y)xdx + (3;^^ — y^)ydy = o,
(x' +- yy = f(/ - x^).
22 EQUATIONS- OF FIRST ORDER AND DEGREE. [Art. 2^.
III.
Exact Differential Equations.
25. An exact differential containing two variables is an
expression which may arise from the differentiation of a func-
tion of X and y. Denoting the function by u, we have
du ■= — dx -\- ■ — ' dy, (i)
dx dy
where the coefficients of dx and dy are the partial derivatives
of u. Thus, the form of an exact differential is Mdx + Ndy.
But, if M and N are any given functions of x and y, we cannot
generally put
du = Mdx + Ndy; (2)
for, if
Jf = ^, and iV = ^, (3)
dx dy
we must have
dM dN , .
^ = 1^' ^*>
because each member of this equation is an expression for .
dxdy
Hence equation (4) is a necessary condition of the possibility
of equation (2) or equations (3) ; that is, of the existence of a-
function whose partial derivatives with respect to x and y are
M and N respectively.
26. It is also a sufficient condition ; for the most general
form of the function whose derivative with respect to x is M, is
u
= JMdx + y, (s)
where Mdx is integrated as if y were constant, and F is a
quantity independent of x, but which may be a function of y.
§111.] EXACT DIFFERENTIAL EQUATIONS. 23
Now the only other condition to be satisfied is that the
derivative of u with respect to y shall equal N; that is,
or
dy\
dy
dy
d
dy.
Mdx
(6)
Since Y is to be a function of y only, but is otherwise
unrestricted, this equation merely requires that the second
member should be independent of x. This will be true if its
derivative with respect to x vanishes ; that is to say, if
dN dM ^
dx dy
This equation is identical with equation (4), which is, therefore,
a sufficient, as well as a necessary, condition.
27. An equation in which an exact differential is equated to
zero is called an exact differential equation. Using the notation
of the preceding articles, the complete integral of the equation
Mdx + Ndy = o when exact is evidently
u ■= C.
The function u is determined by direct integration as indicated
in equations (5) and (6). It is to be noticed that dY consists
of those terms in Ndy which do not involve x ; and evidently
the integral of these terms, and also of those containing x
only, may be considered separately, and it is only necessary to
ascertain whether the terms containing both x and y form an
exact differential. For example, in the equation
x(x + 2y)dx + (x^ — y^)dy = o.
24 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 2"].
the sum of these terms is 2xydx + x'^dy, which is the differen-
tial of x'y ; hence the complete integral is
\xi + x'y — \y^ = C,
or
XT' + 2>x'y — yi = c.
28. An expression involving only some function of x and
y, and the differential of this function, is obviously an exact
differential. Thus, in the equation
xdx + ydy . ydx — xdy _
y'(x^ + y^ — 1) x^ -\- y^
the first term is a function of x^ -\- y^ and its differential, and the
second is a function of - and its differential. The equation may,
in fact, be written
2\J{X- + f - X)^ /xV _^ ^
hence the integral is
o;
y/(^ + y _ i) + tan-'^ = C.
Integrating Factors.
29. We have seen in the preceding articles that tbe com-
plete integral of an ' exact differential equation appears in the
form
u=^C, (i)
so that the differential equation results directly from the differ-
entiation of the integral, C disappearing by differentiation.
§ III.] INTEGRATING FACTORS. 25
Now, since the integral of any equation can be put in the
form (i) by solving it for C, it follows that, whenever we can
solve an equation of the form
Mdx + Ndy = o, (2)
we can produce an exact differential equation which is equiva-
lent to the given equation, that is to say, which is satisfied by
the same simultaneous values of x, y, and p. This new differ-
ential equation being of the first order and degree, must then
be of the form
lj.{Mdx + Ndy) = 0, (3)
where /i is a factor containing x or yor both, but not containing
P-
The factor fx, which converts a given differential equation
into an exact differ-ential equation, is called an integrating
factor.
For example, solving equation (2) of Art. 2 1 for c, we have
c = y -\- -;
y
whence, differentiating,
2xydx — x^dy 2xydx + {y^ — x^)dy
= dy + = y >
and, comparing this with equation (i) of Art. 21, we see that
— is an integrating factor of that equation.
30. A differential equation has a variety of integrating
factors corresponding to different forms of the complete inte-
gral. For example, one integrating factor of equation (i),
26 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 30.
Art. 17, is the factor by means of which we separated the
variables ; namely,
I
(I +^)(i - y)'
and this corresponds to the form (3) of the integral ; but, if we
differenliate the integral in the form (5), Art. 18, we obtain
equation (i) multiplied by the integrating factor
(I - yy
The forms of the integral differ in respect to the constants
which they contain. In general, \i u =. c \i, zxi integral giving
the integrating factor ju, so that
du = fi.{Mdx + Ndy),
then
f{u) = C
where C = f(c) is another form of the integral ; and this gives
the exact differential equation
/'(u)du = o,
or
f(u)fi{Mdx + Ndy) = o.
Hence /'(w)/a is also an integrating factor ; and, since /denotes
an arbitrary function, /' is also arbitrary ; thus, the number of
integrating factors is unlimited.
31- The form of the given differential equation sometimes
suggests an integrating factor. For example, in the equation
{y + logx)dx — xdy = o,
§111.] INTEGRATING FACTORS. 2/
the terms containing both x and y are
ydx — xdy.
This expression becomes an exact differential when divided
either by f- or by x^. The remaining term contains x only ;
hence — is an integrating factor. Thus, we write
ydx — xdy Xo^xdx
x^ x^ ~ '
whence, integrating,
—y log X I
X X X ~ '
or
ex + y + log^ + I = o.
32. The expression ydx — xdy, which occurs in the preced-
ing article, is a special case of a more general one which should
be noticed. For, since
d(x'"y'') = x^-'^ y^-'^{mydx + nxdy'),
an expression of the form
x'^y^{mydx + nxdy) (i)
has the integrating factor
^in — 1— ttyw— I— j3 .
and since, by Art. 30, the product of this by any function of ti,
where u = x'^j/", is also an integrating factor, we have the more
general expression
(in which k may have any value) for an integrating factor.
28 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 32.
As an illustration, take the equation
jv(jy3 + 2x^^dx + x{x^ — 2y'^)dy = o.
This may be written in the form
yi(ydx — 2xdy) + x'^{2ydx + JC^) = o,
in which each term is of the form (i). In the first term,
m = I, n = —2, a = o, yS = 3 ;
and the expression (2) gives, for the integrating factor,
that is to say, any multiplier of this form will convert the first
term into an exact differential. In like manner, any expression
of the form
X^^— Si)*'— I
is an integrating factor of the second term. A quantity which
is at once of each of these forms will therefore be an integrating
factor of the given equation. Equating the exponents of x,
and also the exponents of y, in the two expressions, we have
k — I = 2y4' — 5,
— 2/4 — 4 = U — 1,
from which /& = — 2, and the integrating factor is x~^.
Multiplying the given equation by x-^, we have
y'^x-''' dx — 2y^x-^dy + 2xydx + x^dy = o ;
§ III.] EXAMPLES. 29
and, integrating,
^ + ^^ = ^'
or
20<^y — yt = esc'.
Examples III.
Solve the following differential equations : —
1. {pi' — 4xy — 2y')dx + {f — 4xy — 2x')dy = o,
x3 ^ yi — 6xy{x + y)^= c.
2. -^ = J- ^, ' xy^ = xi'y + c.
dx X 2,y^ — X
3. {2x — y + i)dx + (2J;' — X — i^dy = o,
^-^ — XV + jC^ + jc — J) = ^.
4. jc(a;^ + 3J'^)(/.r + y{y^ + S^)*^ = o,
X* + 6^y + J)J4 = ^.
X(^ — ydx x' + y y
5. ydy + a;^.« + ^2 ^ y = o> ^ 1" ^an-'- = c.
6. (y — x)dy + j'i^ =0, logj + - = <:.
' -^ flic y'^-j — ayi = X + I, yi = ce-'x _x + \ i
dx
a a'
21- x-^ + y = flogx, i = loga; + I + ex.
y
^ , xy _ ax
dy 1 I
dx xy + x^y^ x
v.] DECOMPOSABLE EQUATIONS. 37
CHAPTER III,
EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE.
V.
Decomposable Equations.
38. A differential equation of the first order is a relation
between x, y, and /. If the equation is not of the first degree
with respect to /, the first step in the solution is usually to
solve the equation for /. Suppose, in the first place, that the
equation is a quadratic in / ; then two values of p in terms of
X and y are found. These will generally be irrational functions
of X and y ; in the exceptional case when they are rational
functions, the equation will be decomposable into two equations
^of the first degree. For example, the equation
\dxl
+ y)^ + xy = o (i)
ax
may be written
\dx )\dx I
and is satisfied by putting
L
dx
- y = o (3)
^-^ = (2)
or
dx
38 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 38.
The integrals of these equations are
2_y = x^ + ' — X = u. (2)
If we give particular values to x and y, we find an unlimited
number of values of a differing by multiples of ir; but, writing
the equation in the form
y = tan (x + a),
we see that these values determine but a single curve. We, in
fact, obtain all the curves of the system by allowing a to range
in value from o to tt ; and, as a varies over this range, the curve
sweeps over the whole plane once.
If we take the tangent of each member of equation (2), and
write tan a = ^, we have
y — tan x
= ^,
I + ytanx
in which any simultaneous values of x and y determine a single
value of c ; and c must pass over the range of all real values in
order to make the curve sweep once over the entire plane.
41. In general, if the constant of integration is such that
different values of it always correspond to different curves,
there can be but one value of c for each point ; hence the form
of the integral is
Fc +-Q = o
where P and Q are one-valued functions of x and y, and this
we may regard as the standard form of the integral. It will be
noticed that both P = o and g = o are particular integrals ;
§ v.] SYSTEMS OF INTERSECTING CURVES. 4I
the former corresponding to c = 00, and the latter to t — 0.
Thus, in the example given above, y = tan x and y = — cot x
are particular integrals.
42. In like manner, the form of the differential equation of
the first order and second degree is
Lf + Mp -{■ N = o,
where L, M, and N are one-valued functions of x and y. In
general, two curves, and two only, representing particular
integrals, intersect in a given point. When the expression
Lp^ + Mp + N can be separated into rational factors of the
first degree, these curves belong to distinct systems having no
connection with one another, as in Art. 38 ; but, in the general
case, they are curves of the same system. Thus, the system of
curves representing a proper equation of the second degree is
a system of intersecting curves, two curves of the system, in
general, passing through a given point. Hence, in the integral
equation, given simultaneous values of x and y must generally
determine two values of c, or, at least, values of c which deter-
mine two and only two curves of the system.
43. Take, for example, the equation
f = -i. - y\ (i)
or
dy
= dx.
The integral is
or
±^(1 - f)
sin-'jy — X = a, . ^■- (2)
y = sm{x + a), (3)
in which it is permissible to drop the ambiguous sign, because
jj/ = — sin (;r -I- a) may be written y = sin {x + ir + a), and is
therefore included in the integral (3). Here, as in the example
of Art. 40, if we give particular values to x and y, a has an
42 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 43.
unlimited number of valiies ; for, if 6 be the primary value of
ain-'j)/, every expression included in either of the forms
2«?r + 6 or (2« + l)7r — 6,
where n is an integer, is a value of sin^'x These values of a,
however, determine but two distinct curves, since values of
a differing by a multiple of 27r determine, in (3), the. same curve,
so that each of the above forms determines but one curve.
Equation (3), in fact, represents the system formed by moving-
the curve of sines, y = sin x, in the direction of the axis of x,
and we obtain all the curves of the system while a varies from
o to 277, each branch or wave of the curve falling, when a = 2?r,
upon the original position of an adjacent branch. In this
motion, the curve sweeps twice over that portion of the plane
which lies between the straight lines y = i and j/ r= — i, for
which portion only the value of / is possible in equation (i).
44. If, in the integral of an equation of the second degree,
we so take the constant of integration c, that different values
of it always correspond to different curves of the system, there
can be but two values of c corresponding to a given point. The
equation will then take the form
Fc' +Qc + R = o
where P, Q, and R are one-valued functions of x and y ; and
this may be regarded as the standard form of the integral.
To reduce equation (3) of the preceding article to the stand-
ard form, we have, on expanding,
y = sin X cos a -|- cos x sin a,
in which sin a and cos a are to be expressed in terms of a single
constant. For this purpose, we do not put sin a = c and
cos a = ^(i ~ f)> because this would require us to introduce
§ v.] SINGULAR SOLUTIONS. 43
an irrelevant factor in rationalizing the equation in c ; but we
express sin a and cos a by the rational expressions the sum of
whose squares is unity ; that is, we put
\ — C^ 2C
sin a = , cos a = .
I + '^ — cos^^c) = o,
or
I — J'' = o,
which is identical with the like condition for the /-equation
given in Art. 47.
Cusp-Loci.
49. There are other loci.for points upon which the discrimi-
nants vanish, which it is necessary to distinguish from the
envelope whose equation alone is a singular solution. There
is, in fact, no reason why the values of / derived from the
differential equation, when they become equal as the point
(.r, y) crosses a certain locus, should also become equal to the
value of p for a point moving along that locus. Suppose, then,
that the two arcs of particular integral curves passing through
(.1-, jj/) meet, without touching, the locus for which the values
of p become equal ; and suppose, as will usually be the case,
that the values of / become imaginary as we cross the locus ;
then, when {x, y) is moved up to the locus, the two arcs will
come to have a common tangent ;. and, since they cannot cross
the locus, they will form a cusp, becoming branches of the
same particular integral curve. Thus, the two values of c which
§ v.] CUSP-LOCI. 47
corresponded to the two intersecting arcs will also become
identical, and the locus, which is called a cusp-locus, is one for
which the c-discriminant also vanishes.
For example, the roots of the equation
are equal, each being equal to zero, when
but, since / = oo f or a point moving along this line, this equa-
tion does not satisfy the differential equation. The complete
integral is
in which the condition of equal roots is x'^ = o. The system
of curves is that resulting from moving the semi-cubical par-
abola y^ =. A-3, which has a cusp at the origin, in the direction
of the axis of y. This axis is, therefore, a cusp-locus.
Tac-Loci and Node-Loci.
50. In the preceding article, the values of p were supposed
to become imaginary as we cross the locus for which they
become equal. From equation (2) of Art. 48, it appears that
this will be the case if the discriminant changes sign,* but
otherwise not ; hence, if the factor which vanishes at the locus
appears in the /-discriminant with an even exponent, / will not
become imaginary in crossing the locus. In this case, the two
intersecting arcs cross the locus ; and, when (x, y) is moved up
to the locus, we shall simply have two particular integrals which
touch one another. Such a locus is called a tac-locus. Since
* Since the discriminant is the product of the squares o£ the differences of the
roots, this will be true also for equations of the third and higher degrees.
48 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 50.
the values of c for the two curves remain distinct, the factor
indicating a tac-locus does not appear at all in the ' = i.
* In like manner, the discriminant of a decomposable ^-equation gives a node-
locus. But it is to be noticed that there is no propriety in combining the two
integrals of a decomposable /-equation. Thus, if we combine equations (4) and (5)
of Art. 38, assuming C and c to be identical, we associate each curve of one system ,
with a particular curve of the other system. But if, before doing this, we change
the form of one of the integrals (by introducing a new constant /(<:), as explained
in Art. 30), we associate the curves differently, and obtain a new result, equally
entitled to be considered the integral of the given decomposable differential
equation. '
§ v.] EXAMPLES. 51
4- I ^ ) =0, cey = x^".
\dx I x^
5. f-f- = 4^2, f = c ± \ax.
6. /^ — 5/ + 6 = o, y =■ 2X -\- c, and >» = 3JC + C
8. ^^/^ + 2,xyp + 2jy2 = 0, xy = c, 'and x'^ = C.
9. /3 ^ 2*/^ — y'^p^ — 2xy''p = o,
y =. c, y + x' = c, and jy + i + ^-j = o.
10. /3 _ (;<;2 + ;;^j, ^ y^)p' + XV (j;^ + xy + y^)p — X3j;3 = o,
rv = (fz^j (Ty = I + xc, and 3_y = x3 + '^•
11. /^ + 2/^cotx = y^, J>'(i ± cosjc) = c.
12. /^y - «x3 = o, 25 (j*- - cJ ■= 4«^'.
13. X + xf- = \, y = \/{x — x^) + sin-'v'^ + c.
14. p^(x' + 1)3 =1, iy- cy = -^^.
15. y = {x + i)/%
c^ + 2c(x + I + j) + (^ + I — j)' = o.
16. JV/^ + 2X/ - j; = O, >-' = 2CX + ^^
17. 3X/^ — 6yp + X + 2y = o, c' + c{x — 3jc) + x^ = o.
18. yp + nx = \J{y^ + nx^)sj{i + p"),
52 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 54
19. x^p^ — 2xyp -\- y' = x^y + X*, 2c^ = c^e^ — e-^.
20. zP^y^ ~ ^^yP + 4^^ — a;^ = o,
x^ _|- j;2 _ /^cx + 31^^ = o.
21. ^^(i + /^) = n'ix + j/)%
VI.
Solution by Differentiation.
55. The differentiation of a differential equation of the first
order gives rise to an equation of the second order ; but, in the
cases now to be considered, the result may be regarded as an
equation of the first order, and its integral used in determining
that of the given equation.
Let the given equation be solved for y, that is to say, put
in the form
y = Ax,f); (i)
then the result of differentiation will be of the form
which is of the second order as regards y, but, not containing
y explicitly, is an equation of the first order between x and p.
If, now, we can integrate this equation, we shall have a relation
between x, p, and an arbitrary constant. The result of elimi
§ VI.] SOLUTION BY DIFFERENTIATION. S3
nating / between this equation and equation (i) will therefore
be a relation between x, y, and an arbitrary constant ; hence it
will be the complete integral required.
56. For example, given the equation
-^ + 2xy = x' + y^ ;
dx
solving for y, we have
y =z X -\- \lp; (i)
and, differentiating,
p=^+\f (2)
2y/ dx
Separating the variables x and p, we have
dx =
and, integrating,
dp
2^p{p - i) '
Uog^^^+c,
V/ +
I
or
^ ^_+J^^ (3)
Finally, eliminating / between equations (i) and (3), we have
the complete integral
C + e^^
y = X + ■
g2X
57. In attempting this mode of solution, it will sometimes be
more advantageous to treat y as the independent variable, and
putting / for — , to derive a differential equation involving y
dy
and /. In either case, the success of the method depends
upon our ability to integrate the derived equation. The princi-
54 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 57.
pal cases in which this can be effected are those in which one
of the variables is absent and those in which both variables
occur only in the first degree.
It should be noticed that the final elimination of p is
frequently inconvenient, or even impracticable ; but, when this
is the case, we may express x and y in terms of / which then
serves as an auxiliary variable.
Equations from which One of the Variables is Absent.
58. If an equation of the first order in which x does not
occur explicitly can be solved for /, it takes the directly inte-
grable form
$ = /(;'), (I)
dy
y being treated as the independent variable. Otherwise let it
be solved for y ; thus,
y= <^{p) ; ■ ' (2>
differentiating,
P=4>'iP)% (3)
in which the variables x and / can be separated.
In like manner, an equation not containing y, if not directly
integrable, should be put in the form
X = <^(/).
Differentiating with respect to y, we have
p dy
in which the variables y and / can be separated.
§ VI.] ONE VARIABLE ABSENT. 55
59. As an example, let us take the equation
y = P'-^ y (I)
We have, by differentiation.
/ = (2/ + 2/^)g, (2)
which implies either that
/ = o, (3)
or else that
dx = (2 + 2p)dp (4)
Eliminating p from equation (i) by means of the first of these,
which is not a differential equation for /, we obtain the solution
y = o, (5)
which does not contain an arbitrary constant. But, integrating
equation (4), we have
X ■'r c = 2p + p^,
or
/ = — I + \l{x + c);
and, employing this result to eliminate / from equation (i), we
obtain
y = \-x-c->r\{x-^ (/)> (i)
it may be noticed that
y = 0(o), (2)
(which, since ^ is not necessarily one-valued, may include
several equations) is always a solution, for it gives, by differ-
entiation, / =; o, and thus satisfies equation (i). The reason
of this is readily seen, for the complete integral is capable of
expression in the form
X = >\,{y) + c, (3)
which is the form it would take if derived by direct integration
from the form (i), Art. 58 ; it therefore represents the system
of curves which results from moving the curve
in the direction of the axis of x. If this ctfrve contains points
at which / = o, it is evident that the locus of these points, or
J/ = <^(o), is an envelope ; that is, j/ = ^(o) is a singular solu-
tion.* But, if the point for which / = o is at an infinite
distance, j/ = ^(o) will be the particular integral corresponding
to c = 00 when the integral is written in the form (3). For
* If the /-discriminant were formed, in this case, by the general method (see
Art. 48), we should apparently have (p'(/i) = o as the condition satisfied alike by a
singular solution, a cusp-locus, and a tac-locus. But it is to be noticed, that, when
) is not a one-valued function, the method may fail to detect a case of equal
roots. In fact it is evident, from equation (3), Art. 58, that, if (p'lp) — o, we must
have -/ Of -r| infinite, which indicates a cusp, except when j* = o, which, as we
have seen above, gives a singular solution. Thus, a tac-locus does not satisfy
0'(/). =: o. In the example of Art. 59, the roots of {p) p - (j,{p) ' ■ ■ ■ ■ y^^
which is a linear equation for x regarded as a function of p.
The integral gives ;ir as a function of p ; the elimination of
/ is often impracticable, but, in that case, substituting the
value of X in equation (i), we have x and j/ expressed in
terms of p as an auxiliary variable.
* In this case also, (p'ip) = o determines cusp-loci, but fails to detect a tac-
locus. See the preceding foot-note.
§ VI.] CLAIKAUT'S EQUATION. 59
Clairaufs Equation.
64. The equation
y = px^ f{p), (i)
which is a special case of equation (i) of the preceding article,
is known as Clairaut's equation. The result of differentiation
is
/ = / + ^^ + /(/)?..
ax ax
or
[^ + /'(/)l9 = o.
ax
This equation is satisfied either by putting
X + f{p) = 0, (3)
or by putting
?=o (4)
ax
Equation (3) gives, by the elimination of / from (i), a singular
solution ; and equation (4) gives, by integration,
p = c,
whence, from (i),
y = ^x + f{c) (S)
This is the complete integral, as is verified at sight, since
J> ~ c is the result of its differentiation.
65. The complete integral, in this case, represents a system
of straight lines, and the singular solution a curve to which
these lines are tangent. An example has already been noticed
in Art. 45. Conversely, every system of straight lines repre-
6o EQUATIONS NOT OF THE FIRST DEGREE. [Art. 65.
sented by a general equation containing one arbitrary parameter
gives rise to a differential equation in Clairaut's form, having,
for its singular solution, the equation of the curve to which the
system is tangent. We have only to write the equation in the
form (5), and to substitute / for the symbol denoting the param-
eter. For example, the equation of the tangents to the circle
x^ -\- y^ ■= a?
is
y = mx + ^v'C^ + m^');
hence the differential equation is
y =^ px + asj(\ + p");
or, rationalizing,
{x^ — a'^')p^ — locyp + y^ — a^ = o.
Accordingly the condition of equal roots is found to be
xy — (x^ — «^) {y'' — a") = o, or x' + j/'' = a^.
66. If we form the condition for equal roots in equation (i).
Art. 64, by the general method mentioned in Art. 48, we have
to eliminate p from equation (i) by means of its derivative
with respect to / ; namely,
o = ^ + /'(/),
which is identical with equation (3). In fact, it is obvious that
the condition should be the same ; for, since the complete
integral represents straight lines, there can be neither cusp-
locus nor tac-locus. Precisely the same condition expresses
the equality of roots in the ^-equation, a node-locus being also
impossible.
§ VI.] REDUCTION rO CLAIRAUT'S FORM. 6l
67. A differential equation may be reducible to Clairaut's
form by a more or less obvious transformation. For example,
given the equation
y — 2x^ + ay
dx
(IJ = -
since d{y'^') = 2ydy, if we multiply through by y, y^ may be
made the dependent variable ; thus,
or, putting y"" = v,
_ dv _ a ldv\} .
dx 4 \dxi
hence the integral is
y^ =: ex — \ac^.
Examples VI.
Solve the following differential equations : —
I.
y = -xp + x^f.
X
singular solution, i + /^x^'y = 0.
2.
xf- — 2yp + ax = o,
singular solution, y'' = ax^-
3-
X + py{2f + 3) = 0,
y =
c cp{2p^ + 3)
(I + P')^' '"" (I + P^^
2ap^
X ^(^'-'^ 1 ..
4-
^ ip^ + ^y
(r + ^r
62 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 6/.
5. X -\- yp = af,
'' = ^/(7^-^^' + '''°S[/ + \/(i +/^)]^
6. ;' = (!+ /)x + /% p = ^(^ - ^) + ^^-^'
^ \ -r-F) -^F, |_j, ^ 2 - /^ + f^->(l + /).
7. ;' = «/ + y/Ci + P'),
X — a log [^y + \J{a^ + J(^ — 1)]
+ logCj; - (/(a^ + r - i)] + c.
O i . ^
8. 2JC = JC/ + -,
/
a^c^ — \2acxy + ?>cy^ — i2x'y^ + i6ax^ = o.
9. J)) = a/ + ij>',
X = alog[\/(a' + 4iy) — a] + s/{a^ + /^by) + c.
10. a^j/^ — 4Jf/ + J = o,
c^ + 2fji:(3ay — 8^::^) — 3a'txy + a^f" = o.
11. jy = x/ + v/(^' + «'/'), jv = ex + v'(^' + a'c^),
singular solution, f — = i •
12. (i + x'^')p^ — 2xyp + j/^ — I = o, ji/ = <:;«; + y'(i — ir^).
13- y — Pi^ — l>) -\ — , singular solution, y^ = a^aCx — b).
P
14. ayp'^ + {2x — b)p — y = o, ac^ + c{2x — b) — f = o.
''■ {' - SJ = ^"" + '-'HJ' '' = ^^^ + ^(^ + ^^)-
16. x'(y — px) = yp^, y^ = cx^ + c^.
17. ,
dx a
Let us take as the origin the point of the fixed line on which
the vertex of the parabola falls in the rolling motion. This deter
mines the constant of integration by the condition that jr = o
when / = o, that is to say, when y = a. Integrating, we have
z^{y-
dy — f ^
66 GEOMETRICAL APPLICATIONS. [Art. "JO.
or
y + v/(:i'^ - a") X
log
which may be reduced to the form
^ + e~
a
= a cosh-.
The curve is the catenary.
71. In another class of examples, the curve required is the
singular solution of a differential equation. It is, in this case,
frequently possible to write the complete integral at once, and
to derive the singular solution from it instead of forming the
differential equation. For example, required the curve such
that the sum of the intercepts of its tangents upon the axes
is constant and equal to a. The equation of the curve is
the singular solution of the equation whose complete integral
represents the system of lines having the property mentioned.
The general equation of this system is *
X y
in which c is the arbitrary parameter. Writing it in the form
c^ -f ({y — X — a) + ax = o,
the condition of equal roots is
{y — X — a)' — 4ax = o,
or
(y — x)' — 2a{x + y) + a^ = o,
which is the equation of the required curve, and represents a
parabola touching the axes at the points (a, o) and (o, a).
§ VII.] TRAJECTORIES. 6y
^ Trajectories.
72. A curve which cuts a system of curves at a constant
angle is called a trajectory of the system. The case usually
considered is that of the orthogo7ial trajectory, which cuts the
system of curves at right angles. The differential equation of
the trajectory is readily derived from that of the given system
of curves ; for, at every point of the trajectory, the value of p
has a fixed relation to the value of / corresponding to the same
values of x and y in the equation of the given system of curves.
Denoting the new value of / by p', this relation is, in the case
of the orthogonal trajectories,
p
If, then, we put in place of — in the differential equa-
dy dx
tion of the given system, the result will be the differential
equation of the trajectory. The complete integral of this equa-
tion will represent a system of curves, each of which is an
orthogonal trajectory of the given system. Reciprocally, the
curves of the given .system are the orthogonal trajectories of
the new system.
73. For example, let it be required to determine the orthog-
onal trajectories of the circles which pass through two given
points.
Taking the straight line which passes through the two given
points as the axis of y and the middle point as the origin, and
denoting the distance between the points by 2b, the equation
of the given system of circles is
x" -\- f + ex — b"- = o, (i)
in which c is the arbitrary parameter. The differential equation
derived from this primitive is
{x^ — -f -\- l>^)/fx + 2xydy = o (2)
68 GEOMETRICAL APPLICATIONS. [Art. "J^,-
Substituting — for -f , we have
ay ax
(^2 — ;x^ — i^^dy + 2xydx = o .... (3)
for the differential equation of the trajectories. This equation
is the same as the result of interchanging x and y in equa-
tion (2), except that the sign of b^ is changed ; its integral is
therefore
x^ + y^ +Cy + I?- = o; (4)
and the trajectories form a system of circles having the axis
of X as the common radical axis, but intersecting it and each
other in imaginary points.
74. It is evident that the differential equations of the given
system and of the orthogonal trajectories will always be of the
same degree, and that, wherever two values of / become equal
in the former, the corresponding values of p will be equal in
the latter. Hence the loci of equal roots will be the same
in each case. Now, the trajectories will meet an envelope of
the given system at right angles ; and, since the values of p
become imaginary in both equations as we cross the envelope,
the envelope is a cusp-locus of the system of trajectories.
Conversely, a cusp-locus which is, at each point, perpendicular
to a curve of the given system, becomes an envelope of the
system of trajectories ; but every other cusp-locus is also a
cusp-locus of the trajectories.
In like manner, a tac-locus of the given system becomes a
tac-locus of the trajectories.* A node-locus gives rise to no
peculiarity in the system of trajectories.
* The case in which the tangent curves of the system cross the tac-locus at
right angles forms an exception. In this case, the locus is itself one of the
trajectories ; and being represented, in the common ^discriminant of the two
systems, by a squared factor, we have the case considered in the foot-note on
§ VII.] EXAMPLES. 69
Examples VII.
1. Determine the curve whose subtangent is n times the abscissa
of the point of contact. y- = ex.
2. Determine the curve whose subtangent is constant, and equal
to a, ce^ = v".
3. Determine the curve in which the angle between the radius-
vector and the tangent is one-half the vectorial angle. r=c{\ — cos^).
4. Determine the curve in which the subnormal is proportional to
the «th power of the abscissa. f- = kx"+ ' -\- c.
5. Determine the curve in which the perpendicular upon the
tangent from the foot of the ordinate of the point of contact is constant
and equal to a, determining the constant of integration in such a
manner that the curve shall cut the axis of y at right angles.
The catenary y = a cosh-.
a
page 48. For example, the tac-locus a- =: a in Art. 52 is perpendicular to the system
of curves representing the complete Integral ; the equation of the trajectories is
{x - aYp'' -x^o, (I)
of which the integral is
y ^ C = 2\lx + \la\og '''-^'' (2)
\a + Sx
The system is that which results from moving the curve
a — X
in the direction of the axis of y. This curve is symmetrical to the axis of ^ since
^ X admits of a change of sign, and it has a cusp at the origin, so that the axis of y
is a cusp-locus. The line x = a\s an asymptote which is approached by branches
on both sides of it ; and the result of putting C = 00 in equation (2) is, in fact, this
line, or rather the line doubled, for, if C is infinite, we must, in order to have y
finite, put x — a.
70 GEOMETRICAL APPLICATIONS. [Art. 74.
6. Determine the curve in which the perpendicular from the
origin upon the tangent is equal to the abscissa of the point of contact.
x^ + y^ = 2 ex.
7. Determine the curve such that the area included between the
curve, the axis of x, and an ordinate, is proportional to the ordinate.
8. Determine the curve in which the portion of the axis of x
intercepted between the tangent and the normal is constant, and
interpret the condition of equal roots for /.
2(x — c) = a\og\_a ± \j{a?- — 4^^)] T v/(a^ — Ay')-
9. Determine the curve such that the area between the curve, the
axis of X and two ordinates is proportional to the corresponding arc.
y — cosh ' .
a
10. Determine the curve in which the part of the tangent inter-
cepted by the axes is constant. ^1 +1,1 = ^1.
11. Determine the curve in which a and ^ being the intercepts
upon the axes made by the tangent ma. + «/3 is constant.
The parabola {ny — mxy — 2a{ny + mx) +0^ = 0.
12. Determine the curve in which the area enclosed between the
tangent and the coordinate axes is equal to a^.
The hyperbola 2xy = a^.
13. Determine the curve in which the projection upon the axis
of y of the perpendicular from the origin upon a tangent is constant,
and equal to a. The parabola x'' = ^a{a — y).
14. Determine the curve in which the abscissa is proportional to
the square of the arc measured from the origin.
The cycloid y = asin~'^ + \j{ax — x^).
15. Determine the orthogonal trajectories of the hyperbolas xy — a.
The hyperbolas x^ — y^ = c.
§VII.] EXAMPLES. 71
16. Determine the orthogonal trajectories of the parabolas jc^ = 4«x.
The ellipses 2x^ + y = c'.
17. Determine the orthogonal trajectories of the parabolas of the
nth degree a"''}/ = x". ny^ -\- x' = c'-.
18. Find the orthogonal trajectories of the confocal and coaxial
parabolas y^ = i^a{x + a). The system is self-orthogonal.
19. Show generally that a system of confocal conies is self-
orthogonal.
20. Find the orthogonal trajectories of the ellipses 1- i- = i
when a is constant and b arbitrary. x^ + jc^ = 2a^ log x -\- c.
2 1 . Find the orthogonal trajectories of the cardioids r = a ( i — cos 0) .
r = c{i + cos 6).
22. Determine the orthogonal trajectories of the similar ellipses
— -f ^ = «^, « being the arbitrary parameter. y^^ = cx"^.
a" b^
23. Find the orthogonal trajectories of the ellipses — + ^ = i
when ^ -f - = ^. {xyy^ = «^'+^'.
a" b'^ k"
24. Find the orthogonal trajectories of the system of curves
;-« sin nB = a". r" cos nQ — c".
25. Find the orthogonal trajectories of the curves ;' = log tan Q -\- a.
- = sin^^ -f c.
r
72 EQUATIONS OF THE SECOND ORDER. [Art. 75.
CHAPTER IV.
EQUATIONS OF THE SECOND ORDER.
VIII.
Successive Integration.
75. We have seen, in Chapter I., that the complete integral
of a differential equation of the second order must contain two
arbitrary constants, and that it is the primitive from which
the given differential equation might have been derived by
differentiating twice and using the results to eliminate the
constants. The order in which the differentiations and elimi-
nations take place is evidently immaterial ; for, denoting the
constants by c, and c^, and the first and second derivatives of
yhyp and q, all the equations which can arise in the process
form a consistent system of relations between x, y, c^, c^, p,
and q, and these are equivalent to three independent algebraic
relations between these six quantities. If, after differentiating
the primitive, we eliminate the constant c^, the result will be a
relation between x, y, c^, and p, that is to say, a differential
equation of the first order ; and, if we further differentiate
this equation, and eliminate c„ the result will be the differ-
ential equation of the second order. Now, regarding the latter
as given, the relation between x, y, c„ and p is called a first
integral; and the complete integral, or relation between x, y,
c„ and Cj, is also the complete integral of this first integral, c^
being the constant introduced by the second integration.
§ VIII.] SUCCESSIVE INTEGRATION. 73
76. As an illustration, let the given equation be
g + ^- «
• If this be multiplied by 2p, it becomes
2pJ- 4- 2y-^ = o; (2)
ax ax
and, since this equation is the result of differentiating
/' + r = ^' (3)
(the constant, which is, for convenience, denoted by c^, dis-
appearing in the differentiation), equation (3) is a first integral
of equation (i). It may be written
^- = dx\
and its integral, which is
y
sm-'— = X + a,
c ' '
or
y = £sm{x + a), . . .'. . . . (4)
where a is a second constant of integration, is the complete
integral of equation (i). Expanding sin {x + a), and putting
A = ccosa, B = (Tsina,
^ the complete integral may also be written in the form
y = A%\n.x + Bcosx, (5)
in which A and B are the two arbitrary constants. ,
74 EQUATIONS OF THE SECOND ORDER. [Art. TJ,
The First Integrals.
77. It is shown, in Arts. 14 and 15, that a differential
equation of the second order represents a doubly infinite
system of curves. In fact, if, in the complete integral, we
attribute a fixed value to one of the constants, we have a singly
infinite system ; and, therefore, corresponding to different
values of this constant, we have an unlimited number of such
systems. For example, if, in the complete integral (4) of the
preceding article, we regard f as a fixed constant, the equation
represents a system of equal sinusoids each having the axis
of X for its axis and c for the value of its maximum ordinate,,
but having points of intersection with the axis depending upon
the arbitrary constant a. The first integral (3) is the differ-
ential equation of this system ; and equation (i), which does
not contain c, represents all such systems obtained by varying
the value of c.
On the other hand, if, in equation (4), we regard a as fixed,
we have a system of sinusoids cutting the axis in fixed points,,
but having maximum ordinates depending upon the constant c,
which is now regarded as arbitrary. If now we differentiate
equation (4) and eliminate c, we have the differential equation
of this system, .namely,
y = /tan(x + a), . (6)
which, being a relation between x, y, p and a constant, is
another first integral of equation (i). The result of eliminating
/ between the first integrals (3) and (6) would, of course, be
the complete integral (4).
78. Consider now the form (5) of the complete integral. If
we regard A as fixed, the singly infinite system represented is
one selected in still another manner from the doubly infinite
system ; it consists, in fact, of those members of the doubly
infinite system which pass through the point (|7r, A). The
§ VIII.] THE FIRST INTEGRALS. 75
differential equation of this system, which is fourid by differen-
tiating, and eliminating B, is
jcsinx + pzo%x — A, (7)
which is, accordingly, another first integral of equation (i)
Again, regarding B as fixed, and eliminating A from equation
(5), we obtain the first integral
yco^x — psinx = B (8)
In like manner, to every constant which may be employed
as a parameter in expressing the general equation of the doubly
infinite system of curves there corresponds a first integral of
the differential equation of the second brder. Thus, the
number of first integrals is unlimited.
79. If c, and C2 are two independent parameters, that is to
say, such that one cannot be expressed in terms of the other,
all the other parameters may be expressed in terms of these
two. Accordingly, the two first integrals which correspond to
f, and ^2, which may be put in the form
/.(■«> y, P) = c^, Mx, y, p) = ^2, .
may be regarded as two independent first integrals from which
all the first integrals may be derived. For example, if the first
integrals (7) and (8) of the preceding article be regarded as
the two independent first integrals, equation (3) of Art. 76
may be derived from them by squaring and adding, because
c' = A" -\- B\
It must be remembered that no two first integrals are
independent when regarded as differential equations of the
first order ; for they must both give rise, by differentiation, to
the same equation of the second order. They are only inde-
pendent in the sense that the constants involved are independ
ent, so that they may be regarded as independent algebraic
■Jt EQUATIONS OF THE SECOND ORDER. [Art. 79.
relations between the five quantities x, y, p, c^, and c^, from
which, by the eUmination of /, the relation between x, y, c^,
and c^ can be found independently of the differential relation
between x, y, and p.
Integrating Factors.
80. If a first integral of a given differential equation of the
second order be put in the form f{x, y, p) =. c and differen-
tiated, the result, not containing c, will be a relation between
X, y, p, and q, which is satisfied by every set of simultaneous
values of these quantities which satisfies the given differential
equation. This result will therefore either be the given equa-
tion, or else the product of that equation by a factor which does
not contain q. In the first case, the given equation is said to
be an exact differential equation ; in the latter, the factor which
makes it exact is called an integrating factor. In general, to
every first integral there corresponds an integrating factor.
For example, differentiating equations (7) and (8) of Art. 78,
we find the corresponding integrating factors of the equation
— ^ -j- y = o
to be cos X and sin x respectively. Again, the integrating
factor p was employed, in Art. ^6, in finding the first integral
{3) by means of which we solved the equation.
81. It is to be noticed that an exact equation formed, as in
the case last mentioned, by means of an integrating factor
containing /, is really a decomposable equation consisting of
the given differential equation of the second order and the
differential equation of the first order which results from
putting the integrating factor equal to zero. The exact differ-
ential equation therefore represents, in this case, not only the
doubly infinite system, but also a singly infinite system which
does not satisfy the given differential equation. This system
§ VIII.] INTEGRATING FACTORS. yj
consists of the singular solutions of the several singly infinite
systems represented by the first integral when different values
are given to the constant contained in it. For example, equa-
tion (2), Art. 76, is satisfied by jj/ = C, which does not satisfy
equation (i), but is the solution of / = o ; accordingly, the first
integral (3) has the singular solutions y ^ ±c, which, when c
is arbitrary, form the singly infinite system of straight lines
parallel to the axis of x. In fact, a singular solution of a first
integral represents a line, which, at each of its points, touches
a particular curve of the doubly infinite system. The values
of X, y, and /, for a point moving in such a line, are therefore
the same as for a point moving in a particular integral curve ;
but the values of q are, in general, different ; * hence such a
point does not satisfy the given differential equation.
* The values o£ q will, however, be the same if the line in question has at
every point the same curvature as the particular integral curve which it touches
at that point; and its equation will then be a singular solution. The case is
analogous to that of the singular solution of an equation of the first order; the
giren equation being supposed of a degree higher than the first in q, and a
necessary (but not a sufficient) condition being that two values of q shall become
equal for the values of x, y, and / in question. Suppose, for example, the doubly
infinite system of curves represented by the differential equation to consist of all
the circles whose centres lie upon a fixed curve. In order to determine the
particular integrals which pass through an assumed point (x, y) in the direction
determined by an assumed value of /, we must draw a straight line through (x, y)
perpendicular, to the assumed direction, the required particular integrals being
circles whose centres are the points where this line cuts the fixed curve. These
circles correspond to the several values of q which are consistent with the assumed
values of x, y, and /. When the line touches the fixed curve, two of the values of
q are equal, and the values of x, y, and p satisfy the condition of equal roots in
the differential equation considered as an equation for q. Consider now an involute
of the fixed curve ; its normals touch the given curve ; hence the values of x, y,
and /,at any- of its points, satisfy the condition of equal roots. Now, the circle
corresponding to the twofold value of q is the circle of curvature of the involute,
so that the value of q for a point moving in the involute is the same as its value for
a point moving in a particular integral curve, and the equation of the involute is a
singular solution. Thus the involutes of the fixed curve constitute a singly infinite
system of singular solutions, and the relation between x,y, and p, which is satisfied
JZ EQUATIONS OF THE SECOND ORDER. [Art. 82.
Derivation of the Complete Integral from Two First Integrals.
82. It sometimes happens that it is easier to obtain two
independent first integrals than to effect the integration of one
of the first integrals. The elimination of p between the two
first integrals then gives the complete integral. For example,
as an obvious extension of the results obtained in Art. 80, we
see that both cos ax and sin ax are integrating factors of the
equation
and, since these expressions contain x only, they are also
integrating factors of the more general equation.
g + «^7 = ^ (0
if JT is a function of x only. Thus, we have the exact differ-
ential equation,
co^ax — ^ + a^yco^ax = X cos ax,
dx^
and its integral, which is
X cos axdx + (Ti . . . (2)
dy ,
cos ax -^ + ay sia ax
dx
is a first integral of equation (i). In like manner, the integrat-
ing factor sin ax leads to the first integral
dy
sin ax -^ — ay cos ax =
dx
Xsinaxdx — c^. . . . (3)
by all the involutes (in other words, their differential equation) satisfies the con-
dition of equal roots ; that is to say, it is the result of equating to zero the
discriminant of the ^-equation or one of its factors.
§ VIII.] ELIMINATION OF p FROM TWO FIRST INTEGRALS. 79
Eliminating / between equations (2) and (3), we have
■ay = sin ax X cos axdx — cos ax X sin axdx + c^ sin ax + c^, cos ax,
the complete integral of equation (i).
83. The principle of this method has already been applied
to the solution of equations of the first order in Art. 55. The
method there explained, in fact, consists in forming the equa-
tion of the second order of which the given equation is a first
integral, then finding an independent first integral, and deriving
the complete integral by the elimination of p. But it is to be
noticed that the given equation, containing, as it does, no arbi-
trary constant, is only a particular case of the first integral of
the equation of the second order corresponding to a particular
value of the constant which should be contained in it. Accord-
ingly, the final equation is the result of giving the same par-
ticular value to this constant in the complete integral of the
equation of the second order. For example, in the solution of
Clairaut's equation, Art. 64, the equation of the second order
is — - =: o ; the first integral, of which the given equation is a
special case, is j^ + C = :vp + f{p) ; and the complete inte-
gral is J/ + C = ex + f{c), which represents all straight lines ;
whereas the required result is the singly infinite system of
straight lines corresponding to C" = o.*
* In accordance with Art. 81, it would seem that a singular solution of the
given equation, when it exists, could not satisfy the equation of the second order,
and therefore must correspond to a factor which divides out, just as x +f'(p)
does in the solution of Clairaut's equation. This is indeed true when the
singular solution belongs to the generalized first mtegral, as in this case it does
to_j' -\- C = cx +/(c). But generally the singular solution belongs only to the given
equation ; and there is no reason why a singular solution of a particular first
integral should not satisfy the differential equation of the second order. Thus a
singular solution does nc3t generally present itself in the process of "solution by
differentiation,'' as it does in the case of Clairaut's equation.
8o EQtlATIONS OF THE SECOND ORDER. [Art. 84.
Exact Differential Equations of the Second Order.
84. An exact differential equation of the second order is
the result of differentiating a first integral in the form
f{x, y, p) = c (i)
Hence it will be of the form
in which the partial derivatives -^, -^ and -J-, are functions
ax ay dp
of X, y and / ; so that the latter forms the entire coefficient
of q in the equation. Hence, if a given equation of the second
order is exact, we can, from this coefficient, find, by integration
with respect to p, the form of the function / so far as it
depends upon / ; that is to say, we can find all the terms
of the integral which contain /. These terms being found,
their complete derivative must be subtracted from the first
member of the given differential equation, and the remainder,
which will be a differential expression of the first order, must
be examined. If this remainder is exact, the whole expression
is evidently exact ; and its integral is the sum of the terms
already found and the integral of the remainder.
85. As an illustration, let the given equation be
The terms containing q are (i — x')-£-; and, integrating this
with respect to /, we have (i — x^)p for the part of the integral
VI 1 1. J EXACT EQUATIONS.
which contains /. The complete derivative of this expression
is
ax' ax
and, subtracting this from the first member of equation (i), we
have the remainder
^-r + y = °>
ax
which is the derivative of xy. Hence equation (i) is exact,.
and its integral is
(i - x')p + xy = c, (2>
Again, if we multiply equation (i) by /, it becomes
(i - x')p-£ - xp^ + yp = o (3)
In this form, the integral of the terms containing g is -J(i — x^)p'',
of which the complete derivative is
(i - x^)p^ - xf.
dx
The remainder, in this case, is yp, which is the exact derivative
of \y^ ; hence equation (3) is also exact, and its integral is
(i - ^)/^ + y^ = c^ (4)
Equations (2) and (4) are two first integrals of equation (i) ;
hence, eliminating /, we have the complete integral
c? — 2c,xy + y^ — C2{i — x^) = o, . . . . (5)
which represents a system of conies having their centres at the
origin, and touching the straight lines x ^ ±1.
83 EQUATIONS OF THE SECOND ORDER. [Art. 86.
Equations in which y does not occur.
86. A differential equation of the «th order which does not
contain y is equivalent to an equation of the (« — i)th order
for/. The value of / as a function of x obtained by integrating
this will contain n — i constants ; and the remaining constant
will appear in the final integration, which will take the form
pdx -^ C,
If the given equation is of the first degree with respect to the
derivatives, it will be a linear equation because the coefficients
do not contain y. Thus, if the equation is of the second order,
it may be put in the form
^' + /(.)^ == 0(.),
dx^ dx
or
dx
a linear equation of the first order for /. For example, the
equation
( I + x^) — '^ + x^^ + ax = o
dx^ dx
is equivalent to
dp , x . ax
^t, -I p = .
dx I -{- x' I + xi'
The integral of this is
p = -a +
s/(i + x^y
and, integrating again,
y = c, — ax + c,\og\_x + y/(i + x')'].
§ VIII.] EQUATIONS IN WHICH X DOES NOT OCCUR. 83
87. In general, an equation of the «th order which does not
contain y, and in which the lowest derivative is of the rth
order, is equivalent to an equation of the {n — r)th order for
the determination of this derivative. For example,
d''y , d'^y
dx"" dx^
is equivalent to
— I = a^q.
dx^
Integrating, we have
dx^
and, integrating twice more,
;; = ^f"^ + Be-"^ + Cx + D.
Equations in which x does not occur.
88. An equation of the second order in which x does not
occur may be reduced to an equation of the first order between
y and / by putting
d^y _ d^ _ dl dy _ ,d^
dx'^ dx dy dx dy
For example, the equation
d^^ldy^ (,)
^ dx"- \dx)
thus becomes
//f = /3, (2)
dy
or
d^ _ dy.
84 EQUATIONS OF THE SECOND ORDER. [Art. 88.
whence
/ y
or
dx = -^ -^ c^dy;
y
and, integrating again,
X = logy + £,y + c^ (3)
In equation (2), we rejected the solution p =. o, which
gives J)/ =r C ; but it is to be noticed that the equation is still
satisfied by / = o after the rejection of the factor p; accord-
ingly,";/ :^ C is a particular system of integrals included in the
complete integral (3), as will be seen by writing the latter in
the form
y = A -\- B{x - log;'),
and making B ^= o.
89. If the equation contains higher derivatives, they may,
in like manner, be expressed in terms of derivatives of / with
respect to jr. Thus,
dxT' dx dx^ dy\ dyj dy^ \^y)
In like manner, the expression for the fourth derivative may be
found by applying the operation p — to this last result, and so
ay
on.
The Method of, Variation of Parameters,
90. When the solution of an equation in which the second
member is zero is known in the form y = f(x), the more
general equation in which the second member is a function
of X may sometimes be solved by assuming the value of y in
§ VIII.] THE METHOD OF VARIATION OF PARAMETERS. 85
the same form as that which satisfies the simpler equation,
except that the constants or parameters in that solution are
now assumed to be variables. By substituting for y in the
given equation its assumed value, we obtain an equation which
must be satisfied by these new variables. When the given
equation is of the first order, there is but one new variable,
and the method amounts merely to a transformation of the
dependent variable ; but when the equation is of the «th order,
the assumption involves n new variables, and we are at liberty
to impose n — i other conditions upon them beside the con-
dition that the given equation shall be satisfied. The condi-
tions which produce the simplest result are that the derivatives
of J, of all orders lower than the «th, shall have the same values
when the parameters are variable as when they are constant.
91. For example, given the equation
2+^^-^ = ^' (^>
we assume
y = Cicosff.* + CjSinaA:, (2)
which, if (7, and C-^ are constant, satisfies the equation when
X =■ o. Now, if C, and C^ are variable, we may assume this
value of y to satisfy equation (i), and, at the same time, impose
a second condition upon the two new variables. Differentiating,
we have
-^ = —aCi'sva.ax + aC2 cos ax -\ cos ax -\ sin ax,
dx dx dx
in which the first two terms form the value of — when C, and
dx
C2 are constant. We now assume, as the second condition
mentioned above,
— -co%ax -I -%\-a.ax = 0, (3)
dx dx
86
EQUATIONS OF THE SECOND ORDER. [Art. 9I.
which makes
dy
dx
= — aCiSinajc + aC^cQi.ax.
Differentiating again, we have
d^y , ^ , /-. • dC, ■ , dC,
-^ — —a^C^coiax — a'C^sm.ax — a — -sinajc + a—^coiax.
dx
dx
dx
Substituting in equation (i), we obtain
dC, ■ , dC, V-
a — ■ sm ax + a — - cos ax =^ X
dx dx
(4)
as the condition that y, in equation (2), shall satisfy the given
equation. Equations (3) and (4) give, by elimination.
— a — - = Xiwiax, a — - = Xco%ax;
dx dx
whence
C, =
X sin axdx + c„ C^ =
X cos axdx + c^ ;
and, substituting in equation (2),
y = — zo^ax
I (■
X sin axdx -\ — sin a:*: JSf cos axdx
+ c^ cos ax + c^ sin ax,
as otherwise found in Art. 82.
The method of variation of parameters is of historic interest
as one of the earliest general methods employed. It may
occasionally be applied also when the term neglected in finding
the form to be assumed for the value of y is not a mere
function of x ; but, for the most part, examples which can
be solved by it can be more readily solved by the methods
given in the succeeding chapters.
§ VIII.] EXAMPLES. 87
Examples VIII.
Solve the following differential equations : —
1. — ^ = xe'', y = (x — 2)6^ + o], ax ■'r by = Ae^'' + ^?^ + ^zi?-^.
24. (a3_,3)g_^|' + £! = ,,
a;«r ^ flic a
2a
26. — — + w =
^(9== (i+^^sin'e)i
= J^-^^ ■ ; '- + eco%{B — a).
27. Determine the curve in which the normal is equal to the
radius of curvature, but in the opposite direction.
The catenary jc = ircosh— .
c
28. Determine the curve in which the radius of curvature is
double the normal, and in the same direction.
The cycloid —x = ^sin-' + \l{2cy — y^').
90 EQUATIONS OF THE SECOND ORDER. [Art. 9I,
29. Determine the curve in which the radius of curvature is
double the normal, and in the opposite direction.
The parabola x^ = e,c(^y — c).
30. Show that the equation
dxP- dx
m = o
\dx)
can be solved in the following cases : (a) when P and Q are functions
of X ; (^) when P and Q are functions of y ; (y) when P is a. func-
tion of X and Q a function of y.
In the case (a), the equation is of the "extended linear form,"
Art. 37, for ^ ; in the case (/3), x does not occur, as in Art. 88 ; and in
dx
the case (y), the equation is exact when divided by — .
dx
variation of parameters, the assumed form of -=^ being derived by
dx
In the last case, the equation may also be solved by the method of
iation of parameters, the assumed
neglecting the last term ; the result is
y^'^dy = A^e~l''''^dx + B.
§ IX.] PROPERTIES OF THE LINEAR EQUATION. 91
CHAPTER V.
LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS.
IX.
Properties of the Linear Equation.
92. A linear differentig.! equation is an equation of the first
degree with respect to y and its derivatives. The Hnear
equation of the «th order may therefore be written in the form
dx" (2i«r«-'
in which the coefficients /"<,, P^ . . . P„ may either be constants
or functions of x, and the second member X is generally a
function of x.
We have occasion to consider solutions of linear equations
only in the form j> = f(x), and it is convenient to call a value
of y in terms of x which satisfies the equation an integral of
the equation. Thus, if y, is a function of x, such that y ^ y^
satisfies equation (i), we shall speak of the function y^, rather
than of the equation y =:y,, as an integral of equation (i).
93. The solution of equation (i), whether the coefficients be
variable or constant, is intimately connected with that of
which differs from it only in having zero for its second member.
92 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 93.
Let J/, be an integral of equation (2) ; then C",_y„ where C,
is an arbitrary constant, is also an integral. For, if we put
y = Cj/j in the first member, the result is the product by C", of
the result of substituting jj/ = j/, ; and, since the latter result
vanishes, the former will also vanish.
Again, let y., be another integral of equation (2), which is
not of the form C,j/, ; then will C^^y^^ be an integral, and
C^yi + CiVi will also be an integral. For the result of putting
y = C^i + C2J2 in the first member will be the sum of the
results of putting y = C,_y, and y = Qy^ respectively, and
will therefore vanish. In like manner, if y„ y^, y^ . ■ ■ yn are
n distinct integrals of equation (2),
y = C,y, + C^y^ + . . . + Cnyn .... (3)
will satisfy the equation ; and, since this expression contains
n arbitrary constants, it will be the complete integral of
equation (2). Thus the complete integral is known when n
particular integrals are known, provided they are distinct ; that
is to say, such that no one can be expressed as a sum of
multiples of the others.
94. Now let Y denote a particular integral of the more
general equation (i), and let u denote the second member of
equation (3), that is to say, the complete integral of equation (2).
If we substitute
y = Y + u (4)
in the first member of equation (i), the result will be the sum
of the results of putting y ■=. Y, and y ^=i u respectively. The
first of these results will be X because F satisfies equation (i),
the second result will be zero because u satisfies equation (2) ;
hence the entire result will be X, and equation (4) is an integral
of equation (i). Moreover, it is the complete integral because
71 contains n arbitrary constants. Thus the complete integral
§ IX.] PROPERTIES OF THE LINEAR EQUATION. 93
of equation (i) is known when any one particular integral is
known, together with the complete integral of equation (2).
In equation (4), Y is called the particular integral, and n
is called the complementary function. The particular integral
contains no arbitrary constants, and any two particular integrals
may differ by any multiples of one or more terms belonging to
the complementary function.
Linear Equations with Constant Coefficients and Second Member Zero.
95. In the equation
A^-p- ^ A^'tzll + ... ^ A._.^ + A„y = o, . (i)
dx" dx"-^ dx
in which the coefficients A^, A, . . . A„ are constants, let us
substitute j/ = e""' where m is a constant to be determined.
d d^
Since — e^'' = me'^'', — e""' = w^^""-^, etc. ; the result, after
dx dx^
rejecting the factor e'"'', is
A^m" + ^,OT«-' + . . . + A„_im. + A„ = o, . . (2)
an equation of the «th degree to determine in. Hence, if m
satisfies equation (2), e'"^ is an integral of equation (i) ; and, if
m^, 7n^ . . . m^ are n distinct roots of equation (2),
y = C,e'"^^ + C;^'«2^ + . . . + C„e^n^ • • • (3)
is, by Art. 93, the complete integral of equation (i).
For example, let the given equation be
d^y dy
— ^ i — 2y = o;
dx^ dx
94 LINEAR EQUATIONS : CONSTANT COEFFICIENTS. [Art. 95.
the equation to determine m is
m^ — m — 2 ■= o,
whose roots are — i and 2 ; therefore the complete integral is
y — C,e-^ + (72^==^.
96. Denoting the symbol — by D, equation (i) of Art. 95
may be written
{AoD" + A.D"-- + . . . + An-^D + A„)y = o,
or, symbolically,
A-D)y = o, (I)
in which / denotes a rational integral function. With this
notation, equation (2) of the preceding article becomes
/(ot) = o ;
and, denoting its roots, as before, by m„ m^ . . . m„, equation
(i) may, in accordance with the principles of commutative and
distributive operations (Diff. Calc, Art. 406 et seq), be written
in the form
{D - »^) (^D - m^) . . .{D - m„)y = o. . . . (2)
This form of the equation shows that it is satisfied by each
of the values of j/ which separately satisfy the equations
(I) — m,)y = 0, (i? — m2)y = o„ ... {-D — ■m„)y = o ;
that is to say, by each of the terms of the complete integral.
§ IX.] CASE OF EQUAL ROOTS. 95
Thus the example given in the preceding article may be
written
(Z)+ i)Ci?- 2)y = o,
and the separate terms of the complete integral are the
integrals of
(D + i)y = o and (Z> — 2)7 = o,
which are Qe-^ and C^e""^ respectively.
Case of Equal Roots.
97. When two or more roots of the equation f{m) = b
are equal, the general solution, equation (3), Art. 95, fails to
represent the complete integral ; for, if m^ = m^, the corre-
sponding terms reduce to
in which C, + Cj is equivalent to a single arbitrary constant.
It is necessary then to obtain another particular integral ;
namely, a particular integral of
{D - m,yy = 0, (i)
in addition to that which also satisfies (D — m^y = o.
This integral is obviously the solution of
(Z> — m,)y = Ae»'i'' ; (2)
for, if we apply the operation I) — m, to both members of this
equation, we obtain equation (i). Equation (2) is a linear
equation of the first order, and its complete integral is
"J =
Adx = Ax + B,
96 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 97.
or
y = e'^^^^Ax + B) (3)
Hence the terms of the integral of f(P)y =■ o corresponding
to a double root of f(in) = o are found by replacing the
constant of integration by Ax + B. For example, given
the equation
^_' 2^ + ^ = 0,
dx^ dx^ dx
or
D{D - i)y = o,
the roots of /{m) = o are o, i, i, and the complete integral is
y — C + e^{Ax + B).
98. -If there be three roots equal to m„ we have, in like
manner, to solve
{D — m,Yy = o (i)
But the integral of this is the same as that of
(Z> — m,)y = e>"i^(Ax + B) ; (2)
for, by the preceding article, if the operation (D — m^Y be
applied to each member of this equation, the result will be
{D — mfy = o. The integral of equation (2) is
e-'^i^y = (Ax + B)dx = \Ax^ + Bx + C ;
or, writing A in place of ^A,
y = e"'i'<'(Ax' + Bx + C) (3)
Hence the terras corresponding to a triple root of f(m) = o
are found by replacing the constant of integration by the
§ IX.] CASE OF IMAGINARY ROOTS. 97
expression Ax^ + Bx + C. In like manner, we may show that
the terms corresponding to an ^-fold root m^ are
e'"i^{Ax'--^ + Bx''-'' + . . . + L).
In particular, if the ^-fold root is zero, we have for the integral
of
— ^ = o,
dx'-
y = Ax'--' + Bx'--' + . . . + Z,
as immediately verified by successive integration.
Case of Imaginary Roots.
99. When the equation f{m) = o has a pair of imaginary
roots, the corresponding terms in the complete integral, as
given by the general expression, take an imaginary form ; but,
assuming the corresponding constants of integration to be also
imaginary,* the pair of terms is readily reduced to a real form.
Thus, if m^ ^z a + zj3 and ;«2 = a — 2/3, the terms in question .
are
C^g(a. + {?)^ _|. Qg(.-!^-):^ _ g„jc(^c,e'^'' + Qe-'^^).. . (i)
Separating the real and imaginary parts of e'^'^ and e - '^^, the
expression becomes
(""[(C, + C^)cos/3x + t{C, - OsinySa:];
or, putting C, + Q = A and ^(C, — C^) = B,
e''''{Acosfix + Bsin/3x), (2)
where, in order that A and B may be real, C, and Q ,in (i)
must be assumed imaginary.
98 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 99.
As an example, let the given equation be
{D^ + D + ^)y = o;
the roots are —\ ± \i\ll ; here a = — 1-, /3 = -iy/3 ; hence the
complete integral is
y = e-'^AAco?, — X + -osin — x\.
100. If the equation /(w) = o has a pair of imaginary
r-fold roots, we must, by Art. 98, replace each of the arbitrary
constants in expression (i) by a polynomial of the (r — i)th
degree ; whence it readily follows that we must, in like manner,
replace the constants in expression (2) by similar polynomials.
Thus the equation
d'-y . d^y , _
dx* dx'
or
(i> + lYy = o,
in which ± i are double roots, has for its integral
y — (^A, + .5,,x)cosjc + {A2 + ^2«)sin.ar.
Tke Linear Equation with Constant Coefficients and Second Member
a Function of x.
loi. In accordance with the symbolic notation, the value of
y which satisfies the equation
AD)y = X (I)
is denoted by
y =^ —^X. (2I
§ IX.] THE INVERSE OPERATIVE SYMBOL. 99
Substituting this expression in equation (i), we have
AD)—^—X = X,
^ 'AD)
which may be regarded as defining the inverse symbol (2), so
that it denotes any function of X which, when operated upon
by the direct symbol f{D), produces the given function X.
Then, by Art. 94, the complete integral of equation (i) is
the sum of any legitimate value of the inverse symbol and the
complementary function or complete integral of
f{D)y = o.
This last function, which is found by the methods explained in
the preceding articles, we may call the complementary function
for f{D) ; and we see that two legitimate values of the symbol
— ^— X may differ by an arbitrary multiple of any term in the
complementary function ior f{D) ; just as two values of Xdx
or —X may differ by an arbitrary constant, which is the com-
plementary function for D.
102. With this understanding of the indefinite character of
the inverse symbols, it is evident that an equation involving
such symbols is admissible, provided only it is reducible to an
identity by performing the necessary direct operations upon
each member. It follows that the inverse symbols may be
transformed exactly as if th-ey represented algebraic quantities ;
for, owing to the commutative and distributive character of the
direct operations, the process of verifying the equation is
precisely the same whether it be regarded as symbolic or
algebraic. For example, to verify the symbolic identity
I X = ^i—^ X ^—X
D^ — a' 2a\D — a D + a
lOO LINEAR EQUATIONS: CONSTANT COEFFICIENTS.{Art. 102.
we perform the operation D^ — a^ on both members ; thus
X = ±\{D + a){D- a)-l—X- {D - a) (Z? + d)—^x'\
2a\_ D — a D + a J
= J-[(_D + a)X - (D - a)X~\ = —2aX = X,
2a\_ J 2a
the process being equivalent to that of verifying the equation
-L/_J L_\
2a\D — a D + aj
Z)^ - fl^
considered as an algebraic identity.
103. The symbol X denotes the value of y in the
D — a
equation of the first order
^-ay = X;
dx
hence, solving, we have
■ X = e''Ae-''''Xdx.
D — a
By repeated application of this formula, we have
(0
^ -X = — ^ — e'^'^
{D - ay D - a
and, in general,
e-^^Xdx = f"-^
e-a^Xdxdx; (2)
■X = e"''
. . . le-^^Xdx^, .... (3)
in - ay
the last expression involving an integral of the rth order.
§ IX.] GENERAL EXPRESSION FOR THE INTEGRAL. lOI
General Expression for the Integral.
104. We may, by means of equation (i) of the preceding
article, write an expression for the complete integral of
f{D)y = X involving a multiple integral of the n\h. order.
For, using the notation of preceding articles, we may put
f{D) = {D - m,){D - m,)...{D - m„);
whence
I Y - ''
I
D - mn
/{£>) Z> - m,£) - m/ '
= e'''^-'
g(,m2-mi)x
e-"'nXdx";
but the expression given below is preferable, involving, as it
does, multiple integrals only when the equation f{D) = o has
multiple roots.
105. Let be resolved into partial fractions ; supposing
m^, m^ . . . m„ to be all different, the result will be of the form
\ N N TV
AD) D - m, D - m^ D - m„
in which N^, N^ . . . N^ are determinate constants ; hence, by
equation (i). Art. 103,
^^X = N,e'»i^\e-'"i^Xdx + . . . + N^e'^nx
{D) J -r . -r «
e-i'hi^Xdx, (2)
AD)
which is the general expression * for the complete integral
* First published by Lobatto, " Theorie des Caracteristiques," Amsterdam,
1837; independently discovered by Boole, Cambridge Math. Journal, ist series,
vol. ii. p. 114.
I02 LINEAR EQUATIONS: CONSTANT COEFFICIENTS.\Kx\.. 105.
when the roots of /(Z>) = o are all different ; each term, it will
be noticed, containing one term of the complementary function.
When two of the roots of /(/?) = o are equal, say 7«, = m^,
the corresponding partial fractions in equation (i) must be
assumed in the form
D ^ m, {D — tn,y '
and then by equations (i) and (2), Art. 103, the corresponding
terms in equation (2) will be
JV^gm^x
e-'ti^Xdx + N:,e'"i^
"Xdxdx.
In like manner, a multiple root of the rth order gives rise to
multiple integrals of the ^-th and lower orders.
106. When f(P) = o has a pair of imaginary roots, u ± ?/3,
we may first determine, for the corresponding quadratic factor,
a partial fraction of the form
{D - ay + /3^
The corresponding part of the integral will be found by applying
the operation N^D + N^ to the value of
-X.
{D - ay + /3-
Decomposing the symbolic operator further, this expression
becomes
I / I I Vy.
2«73\Z» - a - «•/? D - a + ip) '
that is,
J_ g(a. + 0)x\g-(,a + 0)xXdx —e(-'^-'^)Ae~<^<^-'^)xXdx
zip J 2//3 J
§ IX.] EXAMPLES. 103
This last expression is the sum of two terms of which the
second is the same as the first with the sign of i changed ;
and, the first term being a complex quantity of the form
P + iQ where P and Q are real, the sum is 2P, or twice the
real part of the first term. Hence
e°^ r
= therealpartof -— (cos/S.;c + i%va.^x) i?-"(cos^^ — i%\u^x)Xdx,
. ip J
or
X
{D - ay + ^
e^ sin j8;
y8
e-'^smPxXdx.
When a = o, this result reduces to that otherwise found in
Arts. 91 and 82.
Examples IX.
Solve the following differential equations : —
I. ^ - 5 T^ + ^J = o, y = c,e^^ + c^ev^.
ax? ax
dx^ ax
3. .*(g +.)=(-+*■)!.
ax bx
I04 LINEAR EQUATIONS CONSTANT COEFFICIENTS.[Arl. Io6.
4. —^ — 2^ + sj = o, y ■=■ e^(Acos 2x + ^sin 2x).
dx^ dx
d^y d'^y
dx^ dx^
y = c^e" + c^e-'^ + A'iva.\{x + a).
dx^ dx^ dx
dx^ dx
d^y .
1 1 . — - + 2 —^ = 2 ^^ + ^,
^/^ «;i;3 dx
y = c,e^ + e-^{c2 + c^x + c^x^).
12. g-4^ + 8j^-8^ + 47=o,
fltn ax' a^^ dx
y = i?-^( - ay + ^e-
= — 3 (asinySx -|- /8cos/3;c)U— "^cos/Sx^iaJx:
— (acosjSx — /3sinjS;c)
if-°-'^sin/S;!irX(/;c
1 7. Show that ■
-^
= — %vaax\cosaxXdx — cos iz;c sin a.r^ir/;c
2«'L J . J .
^ cos «a: cos axXdx? + sin «:r
2«1 J J
sin axXdx^
IS. -^ - 2-/ + 4JV = ^,
in other words, let it be required to solve the
equation
f{D)y = e- (i)
Since, as in Art. 95, Z'''^^ = a''e"'', and f{I)) is a sum of terms
of the form Aiy,
f{P)e^^ = f{a)e^' ; (2)
whence
Here /(«) is a constant ; and therefore, except when /(a) = o,
we may divide by it and write
AD) f{a) '•
which is the value of y in equation (i). Thus we may, when
the operand is of the form Ae"^, put i? = « in the operating
symbol except when the result would introduce an infinite
coefficient.
§ X.] SYMBOLIC METHODS OF INTEGRATION. lO/
io8. In the exceptional case, equation (2), of course, still
holds ; but it reduces to f(D)e^^ = o, and thus only expresses
that e""^ is a term of the complementary function. In this case,
we may still put a for D in all the factors of f(D) except
D — a. Thus, putting
fiD) = (Z? - a)<^{D),
we have
f{D) D - a («)
Again, \i f{D) = {D — aYj>{D), so that a is a double root
of /(/?) = o, we shall have
_r gax _- . z ^ajr __ ^flj:
f{D) iD-ay {a + /i) { 2 ^ J
because, a being a root of /{s) = o, /(s) = {2 — a){z), and
/{a + k) = h<^(a + h).
§ X.] SYMBOLIC METHODS OF INTEGRATION. IO9
Now, making ^ = o in this result, we obtain
AD) ct>ia)
as before.
This is an instance of a general principle of which we shall
hereafter meet other applications ; namely, that, when the par-
ticular integral, as given by a general formula, becomes infinite,
it can be developed into an infinite term which merges into the
complementary function, and a finite part which furnishes a new
particular integral.
Again, when « is a double root, and X = c"^, the infinite
expression can be developed into two infinite terms which
merge into the complementary function, together with a finite
term which gives the new particular integral. For example,
since ft is ultimately to be put equal to zero, we may write
e^^ 1,1, K'x^ ,
I + nx + h
(Z> - ay<^{D) (j){a + h)h-
(f){a + h)h\
The first two terms have infinite coefficients when h — o, but
they belong to the complementary function ; the third term is
finite, and gives the particular integral
, L e^x ^ ^'^"^
{D - ay4>{D) 2(j}{a)'
Case in which X contains a Term of the Form sin ax or cos ax.
III. We have, by differentiation,
D'&va.ax = a cos ax, I^smax = —a' sin ax,
lf''smax= ( — a')''s\nax ;
whence
/(ly) sin ax = /{ — a'') sina^.
I lO LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 1 1 1.
and, in like manner, we obtain
/{D^)co5ax = f{ — a^) CO?, ax.
It follows, as in the similar case of Art. 107, that
sin ax = sin ax, f i ~)
and
_L_cos<^^=-^^cos«^, (^)
except when /(—«'') = o. It is obvious that we may include
both these results in the slightly more general formula
j^^sm{ax + a) = -_2__sin(«^ + a).
For example, to solve
— ^ - ;/ = sin {x + a),
we have, for the particular integral,
^^^^— sin(a: + a) = -|sin(^ + a).
Adding the complementary function, we have the complete
integral
y = c^e^ + c-^e-"^ — ■|sin(jc + a).
112. In order to employ equations (i) and (2) when the
inverse symbol is not a function of Z*^, we reduce it to a
fractional form in which the denominator is a function of
§ X.] SECOND MEMBER OF THE FORM sin ax OR COS aX. Ill
D'. This is readily done ; for we may put /(D) in the form
/,{!)') + D/(J)^), and the product of this hy f,{D^) - Df^{D^) .
will be a function of D^- Moreover, since we have ultimately
to put D^ = —a', we may at once put —a^ in place of D^ in the
expression for flD), which thus becomes
M-a^) + £>M-a^).
For example, given the equation
(Z)^ + -O — 2> = sin 2^;
the particular integral is
I . I . Z> + 6 .
■ sin 2x = sm 2x = sin 2x
I> + D — 2 D — 6 D" - lb
D + b - cos 2x + x sin 2x
= ' — sin 2x = ■ — ^ .
40 20
Adding the complementary function
y = C,e- + C,.-- - ^0S2^ + 3sin2^_
20
113. The case of failure of the formulae (i) and (2) of
Art. 1 1 1 takes place when the operand is a term of the
complementary function. Thus, if the given equation is
-^ + a^y = cos ax,
doc" ^
the complementary function is A cos ax -\- B sin ax. Accord-
ingly, in the particular integral -— ^cosax, the substitution
D^ ^ ~ a^ gives an infinite coefficient. The most convenient
112 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 1 1 3.
method of evaluating in this case is that illustrated in Art. i lo.
Thus, putting a -\- h for a in the operand, and developing
cos (ax -\- hx) by Taylor's theorem,
cos {a + K)x
•(« + hy +
— I cos ax — sm ax . hx — cos ax . !-•••)•
a\ . 2 /
Omitting the first term which belongs to the complementary
function, we may write, for the particular integral,
^ cos (a + h)x = ( X sin ax + — cos ax — . . . ) ;
n^ -^ a^ ' 2a + h\ 2 I
and, making h = o, we obtain
I X sin ax
cos ax =
B' + a^ 2a
and the complete integral of equation (i) is
> , D , X smax
y = A cos ax + B cos ax -\ .
Case in which X contains Terms of the Form x^.
114. If an inverse symbol be developed into a series pro-
ceeding by ascending powers of D, the result of operating upon
a function of x with the transformed symbol is, in general, an
infinite series of functions ; but, when the operand is of the
form x"^, where m is a positive integer, the derivatives above
the mth. vanish, and the result is finite. For example, to solve
-^ + 2y = xi,
ax
§ X.J SECOND MEMBER OF THE FORM X""- II3
the particular integral is
-x^ — -xi
£> + 2 2 I + JZ>
= i(l - ii? + iB^ - 1^)3 + . . .)X3
= i{xi - %X^ + 3^-1);
and the complete integral is
y = O--" + ^X'i — Ix^ + |x — |.
This result is readily verified by performing upon it the opera-
tion D -\- 2.
115. When the denominator of the inverse symbol is divis-
ible by a power of D, the development will commence with a
negative power of D, but no greater number of terms will be
required than would be were the factor D not present. For
example, if the given equation is
(Z)4 + D^ + D^)y = x^ + sx',
the particular integral is
y = (xi + 7X^) = — (xi +' ■zx')
= -^ri - (Z) + £>') + (£> + D'Y -{D-^D-Y^.. ."](x3+ 3^).
Since the operand contains no power of x higher than jr3^ jt is
unnecessary to retain powers of D higher than D^ in the
development of the expression in brackets. Hence we write
^ = :^(^ - ^ + ^')(^' + 3*^) = (^-j,^ ^)(*' + 3-^)
X^ JC* x^
= 1 xT- + ^x^ 4- 6x,
20 4 4
114 L/NEAJi EQUATIONS: CO-NSTANT COEFFICIENTS.[Art. 11$.
in which the last term should be rejected as included in the
complementary function. Thus the complete integral is
_)' = x^ + 2,x^ + c^x + C2 + e-i^l c, cos
20
\ 2 2 y
It will be noticed that, had we retained any higher powers of
D in the final development, they would have produced only
terms included in the complementary function.
Symbolic Formulce of Reduction.
116. The formulas of reduction explained in this and the
following articles apply to cases in which X contains a factor
of a special form.
In the first place, let X be of the form e'^'' V, V being any
function of x. By differentiation,
Agaxy ^ ea.x^ 4. ae'-^'V,
dx dx
or
DC^'V = e^''{D + a)V. (i)
By repeated application of this formula, we have
jr)2gaxy = De'^iD + a)V = e'"'{D + ayV;
and, in general,
£)reaxy — e''^{D + ayV.
Hence, when <^{P) is a direct symbol involving integral powers
of D, we have
(l){n)e'"'F = C^^iU + a)F. (2)
§ X.] SYMBOLIC FORMULAE OF REDUCTION. I15
To show that this formula is applicable also to inverse
symbols, put
whence
V = — L_ V ■
and equation (2) becomes
^{JJ + a)
in which F", denotes any function of x, since V was unrestricted.
Now, applying the operation — — to both members, we have
(j}(D)
_J^ea^y = e^^ 1 y (3)
which is of the same form as equation (2).
As an example of the application of this formula, let the
given equation be
fly
The particular integral is
y = e'^^x = e'^^
I f=^ I
D' + 4D + T, 3 I + |Z> + ^i?^
3 3 9
1 16 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 1 1/,
117. The formula of reduction of the preceding article may
often be used with advantage in the evaluation of an ordinary
integral. For example, to find
formula,
gmx sjjj nxdx, we have, by the
— e'"^ sin nx = e'"^ sin nx :
B D + m
hence
T) — y/i
e'"^ sm nxdx — e"'-'' sm nx
ly — w}
= (m — D) sin nx = ym sm nx — n cos nx) .
tn' + n" m^ + n^
It may be noticed that equation (i), Art. 103, is a case of
the present formula of reduction, for
^ X = ^ — e''^e-'"'X;
D — a D — a
hence, applying the formula, we obtain
-X = e''='—e-''^X = i
D — a D
e-'^^Xdx;
in which we pass from the solution of a differential equation to
a simple integration. In the above example, on the other hand,
we employed the same formula to reverse the process, the direct
solution of the differential equation being, in that case, the
simpler process. Compare Int. Calc, Art. 63.
118. Secondly, let X be of the form xV. By successive
differentiation, we have
DxV = xDV + V,
D^xV = xD^V + 2DV,
DKxV = xD^V +■ iiyV]
and, generally,
ly-xV = xiyv + riy-^v. (i)
§ X.] SYMBOLIC FORMULM OF REDUCTION. WJ
Now let ^(D) denote a rational integral function of D, that is,
the sum of terms of the form a^D'' ; and let us transform each
term of ^(D)xV by means of equation (i). We thus have
two sets of terms whose sums are xla^D^V and ^a^rD^-^V
respectively. The first sum is obviously X(f>(D) V; and, since
a^rD''-^ is the derivative of a^I)'' considered as a function of £>,
the second sum constitutes the function (/>'(/?) V. Hence
^{D)xV = x4>{D)V + '{V)V, .... (2)
where <^' is the derivative of the function ^.
To show that this formula is true also for inverse symbols,
put
whence
^ V .
and equation (2) becomes
or
in which V^ denotes any function of x. Hence, applying the
operation — ^— to both members, we have the general formula
which is of the same form as equation (2), because — 2; is the
9
derivative of the function — .
9
Il8 LINEAR EQUATIONS: CONSTANT COEFFICIENTS.[Art. 1 1 9.
119. As an example, take the linear equation
dy
-^^ — y = x%\nx.
dx
By the formula, the particular integral is
I ■ . I . I .
-jrsm^= X-
Z> - I D - \ {D - lY
Z) + I . D^ + 2D + 1 .
= X ■ sm X — — sm x ;
Z)^ - I {^ - i)"
hence
y = — ■|a:(cos* + sinj;) — ^cosj; + Ce^.
This example is a good illustration of the advantage of
the symbolic method, for the general solution would give the
integral in the very inconvenient form
y = e""
e-^x sin xdx + Ce^ ;
and, in fact, the best way to evaluate the indefinite integral in
this expression is by the symbolic method, as in Art. 117.
120. Finally, let X be of the form x^ V. Putting xV va place
of V in formula (2), Art. 118,
(f>iD)x'F = x:f>{B)xF + ^'{D)xV;
and, reducing by the same formula the expressions {-D)x V and
'{D)xV, this becomes
4>(I))x' V = x^(^(Z>) V + 2X(f>'{£>) V + ^'\D) V. . (4)
Again, putting xV for V in this formula, and reducing as
before, we have
(f,{I))x^V=x^4>{n)V + sx''{I))V+ sx(t)"(D)V+ — 2 .
= x^ sin 2x + 2x sm 2x -\ sm 2X
i> + I {D" + I)' {I> + 1)3
x^ . ?>x , 26 .
= sin 2JC cos 2x ■\ sin 2x,
3 9 27
and the complete integral is
qx^ — 26 . Zx
y =. c^ cos X + ^2 sm jc — 2 sin 2x cos 2x.
27 9
Employment of the Exponential Forms of sin ax and cos ax.
121. It is often useful to substitute for a factor of the form
sin ax or cos ax its exponential value, and then to reduce the
result by means of formula (2) of Art. 116. P'or example, in
solving the equation
-^ + V = a:^ sin *■,
dx" ^
we have, for the particular integral,
y — ^ x's.iax = ^ —(e" — e-");
but it is rather more convenient to write, what is easily seen to
be the same thing, since e" = cos;r + i sin;tr,
y = the coefficient of/ in — x^'e'^.
^ D- + 1 ,
I20 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 121.
Now
I
-x'ei'^ = e'-' x^ = e'^ x^
n^ + 1 {D + iy + I D{D + 2i)
= ( I 1 ... jJC^
21 D\ 21 6,1^ I
V 2/) 4 8 /
COS X -^ l'a\ViX)\ H 1 );
\ 6 4 4/
whence, taking the coefficient of i, and adding the complement-
ary function,
y = cos^(^ — \x'>' + \x) + sinji:(^ + \x^').*
Examples X.
Solve the following differential equations : —
T ^
— y = xe"^^ + if^.
y = c^e'' + C:,c + —{ix — 4) +
xe^
* This method has an obvious advantage over that of Art. 120 when a high
power of X occurs. Moreover, when, as in the present example, the trigonometrical
factor is a term of the complementary function, it should always be employed. For
it is to be noticed that, in formula (3), Art. 118, while two legitimate values of the
symbol in the first member can differ only by multiples of terms in the comple-
mentary function of i^iD], two values of the second member may differ by the
product of one of these terms by x. Hence a result obtained by the formula might
be erroneous with respect to the coefficient of such a term. In the example of
Art. 119, the uncertainty would exist only with respect to a term of the form xe^,
but it is easy to see that no such term can occur in the solution. In the example
of Art. 120, a similar uncertainty exists with respect to terms of the form JT^sin jt,
x^ cos X, X sin x, and x cos x, none of which occur in the solution. In the present
example, if solved by the same method, the uncertainty would exist with respect to
terms of the same form ; and, as such terms do occur in the solution, an error
might arise. See Messenger of Mathematifs^ vol. xvi. p. 86.
X.] EXAMPLES. 121
2. -^ — 2y = x^ + e^' -^ cos 2x,
dx
y = ce'^ — e^ — 5(4^' + 6jf^ + fijc + 3)
+ J(sin2j; — cos2x).
>3j:
y = {A -{■ Bx)e^ + —(2*=' - 4.ac + 3)-
o
d^y , d^y ,
y = (CjX + C2) sin ^ + ((Tj^: + ^4) cos * — J:t:^ sin jc.
5. ^ — 2-^ + JC = xe'^, y = tf^((r. + c^x + |jc3).
6. — =^ + 4_)' = sin 2,x + e^ + x^,
y = Acos2x + ^sin2;«: + |(tf^ — sin 3a:), + l{2x' — i).
d^v dv
7. — ^ — 2 ^- + aj- = 1?^ sin ^ + cos x,
dx^ dx
y = ^-^(^cosa: + ^sin^) — ^x^^cosjc + ^(cos;r — 2sinx).
d'v
8. —^-{-y = xsin 2X,
dx^
y = A cos X + JB sin x — ^x sin 2X — ^ cos 2X.
d^'y ,
Q. — =i + y = xsmx,
^ dx"
X^ X '
y = A cos ;j; + ^ sin ^ cos x -\ sin x.
4 4
10. — + 4y = 2x3 sin^ X,
dx'
y = Asm2X + jB cos 2X H "*
8
8^3 _ -j^ AX* — xx^ .
— ^^ ^ cos 2JC — — sm 2X.
128 64
122 LINEAR EQUATIONS: CONSTANT COEEFICIENTS.[Axt. 121.
1 1 . — =^ — y := e^ cos X,
y = Complementary Function — if^ cos x.
d^y , ■ 1 • 1 cos 2x , X sin .r . ^ „
12. — -^ + j(; = smf^sm|;c, y = — + i.Siiii.'- + c.F.
dxr 120 12
13- r: + 32 ? + 48;' = xe-'^, y = ^^ll^'c^s + x^) + c. F,
fl'x* dx 144
14. — ^ + 2^ + ^ = a^cos^,
x^ sin a: , 0^2 — X* , /-. T.
J" = 1- 2 cos x + C. F.
12 48
15. ^^ — 2-^ + 4J); = ^'-^cosa:,
^ _ ££_(3sina; — cosx) + C. F. (Compare Ex. IX., 18.)
20
16. (— + i\ y = X' + x-\
y = e-^{c^ + C:,x + c^x') + ;c^ — 6ar + 12 + ^-^
17. (-C + 3)'y = cosa:r,
y = e-^^^c, + c^x + . . . + c„x«-^)
— dxi.
+ (a^ + d') 2 cos lax — n cot" ' - |.
18. Expand the integral \x"e^dx by the symbolic method.
—x"e^ = e^lx" — fix"-^ + n{n — i)a;«-=' — . . .J + ...
19. Prove the following extension of Leibnitz' theorem : —
{D)uv = u . 4>{D)v + Du . <^'{D)v + — . ^"{D)v + . . . ,
2 !
and show that it includes the extended form of integration by parts,
Int. Calc, Art. 74.
§ X.] EXAMPLES. 123
20. In the equation connecting the perpendicular upon a tangent
with the radius of curvature,
(Diff. Calc, Art. 349), / and ^ may be regarded as polar coordinates
of the foot of the perpendicular. Hence show that, if the radius of
curvature be given in the form p = /(), the equation of the pedal is
r=bco^{e + o) +— i_/(e),
and interpret the complementary function (W. M. Hicks, Messenger of
Mathematics, vol. vi. p. 95).
2 1 . The radius of curvature of the cycloid being p = 4a cos <^,
find the equation of the pedal at the vertex. r = 2a& sin Q.
124 LINEAR EQUATIONS: VARIABLE COEFFICIENTS.[Axt. 122.
CHAPTER VI.
LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS.
XI.
The Homogeneous Linear Equation.
122. The linear equation
ax" ax" - '
in which the coefficient of each derivative is the product of a
constant and a power of x whose exponent is the index of the
derivative, is called the homogeneous linear equation. The
operation expressed by each term of the first member is such
that, when performed upon x'", the result is a multiple of x"^ ;
hence, if we ^it y ^ x'" \n the first member, the whole result
will be the product of x"" and a constant factor involving m.
Supposing then, in the first place, that the second member is
zero, the equation will be satisfied if the value of m be so taken
as to make the last-mentioned factor vanish. For example, if,
in the equation
-d'^y , dy , ,
X' —^ + 2X ^ — 2y = o, (i)
dx^ dx
we put J = X'", the result is
\^m{m — i) + 2m — 2']x'" = o;
§ XL] THE HOMOGENEOUS LINEAR EQUATION. 1 25
hence, if m satisfies the equation
»?' + »? — 2 = o, (2)
x"" is an integral of the given equation. The roots of equation
(2) are i and —2, giving two distinct integrals ; hence, by
Art. 93,
y = c^x + c^iX-^
is the complete integral of equation (i).
The Operative Symbol &.
«
123. The homogeneous linear equation can be reduced to
the form having constant coefficients by the transformation
X = ^. For, if .*• = (^, we have (Diff. Calc, Art. 417)
d_ ^ d_
' dx dd'
and, in general,
rjiL== 1(1
^;^ = l9(|-0---(|-''+^)
so that in the transformation each term of the first member of
the given equation gives rise to terms involving derivatives with
respect to 6 with constant coefficients only. Denoting — by D,
the equation is thus reduced to the form
/{D)y = (i)
in which/ is an algebraic function having constant coefficients.
126 LINEAR EQUATIONS: VARIABLE COEFFICIENTS.[Art. 1 23.
Now, if we put d- for the operative symbol x-— . the trans-
dx
forming equations become
ax ay?
and, in general,
XT— = %(& - \){& - 2) . . A& - r -^ 1);
and the result of transformation is
f{&)y = 0* (2)
in which f denotes the same function as in equation (i), but x
is still regarded as the independent variable. As an example
of the transformation of an equation to the form (2), equation
(i) of Art. 122 becomes
{%{& - i) + 2& - 2\y = o,
or
({5-2 + ^ _ 2)y = o.
124. The operator & has the same relation to the function
X'" that D has to e""' ; for we have
&x'" = mx'", &'x"' — m^x'", . . . &''x"' = mrx'" ;
whence
f{{)-)x"' = f{m)x"' . . (i)
* The factors x and — of the symbol x — are non-commutative with one
dx dx
another, and the entire symbol, or iJ, is non-commutative both with x and with D ;
but it is commutative with constant factors, and therefore is combined with them in
accordance with the ordinary algebraic laws.
% XL] 7V/£ OPERATIVE SYMBOL ^. 12/
Thus the result of putting y ■=. x"^m the homogeneous linear
equation
/Wj' = o (2)
is f{in)x'" = o ; whence
A'») = o (3)
Accordingly, it will be noticed that the process of finding the
function of in, as illustrated in Art. 122, is precisely the same
as that of finding the function of &, as illustrated in Art. 123.
If, now, the equation /(in) = o has « distinct roots, ;«„ w/^
. . . m„, the complete integral of /{&)jy = o is
y = C.x'"^ + C:,x"'^ + . . . + CnX'"" ; .... (4)
the result being the same as that of substituting x for ^^ in
equation (3), Art. 95.
Cases of Equal and Imaginary Roots.
125. The modifications of the form of the integral, when
fifl) = o has equal roots, or a pair of imaginary roots, may be
derived from the corresponding changes in the case of the
equation with constant coefficients. Thus, when /{&) = o has
a double root equal to ;«, we find, by putting x in place of {&) where — bx){Dy — bxy) — Dy — bxDy — by— bxDy + b^xy,
the equation may be written in the form
{D — bxYy + iJy = o j
or, putting ^ for D — bx,
(C^ + b)y = o,
in which the operator is expressed as a function of ^. Resolving
it into symboHc factors, we have
i^-t\/b){^ + i\/b)y = o;
and the two terms of the integral satisfy respectively the
equations
(^ — t^b)y = o and (f + i^b)y = o.
The first of these equations gives
{D — bx — i^b)yi = o,
or
■^ = (bx + i\lb)dx;
and, integrating,
logjl'i = ^bx^ + isjbx + (Tj,
or
y^ = Cie^^^{co5xi^b + ismxsjb').
In like manner, the second equation gives
y^ = C:^e^^^^{cosx — x'^')y = v;
then the equation becomes
(P — x)v = X,
(4)
(5)
a linear equation of the first order for v. Solving equation (5),
•we have
and, substituting in equation (4), we have, by integration.
y = e^^
,—^Jfi + \k^
e-'^^Xdsc' + c,ei^^
e- J^^ + ^■'^dx + c^e\^'^. . (6)
131. The solution of the general linear equation of the first
order
(£> + P)y = X
may (see Art. 34) be written in the symbolic form
y = ^^^X = e-V'^'^eV'-Xdx,
which includes the complementary function since the integral
sign implies an arbitrary constant. In accordance with the
same notation, the value of y, in equation (2), would be written
y =
D — x" D — X
X,
which is at once reduced to the expression (6) by the above
formula. It will be noticed that the factors must, in the in-
verse symbol, be written in the order inverse to that in which
they occur in the direct symbol.
134 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 1 3 I.
In obtaining this solution, the non-commutative character
of the factors precluded us from a process analogous to the
method of partial fractions. Art. 105 ; we have, in fact, only a
solution analogous to equation (i) of Art. 104.
Examples XI.
Solve the following differential equations : —
^d'iy , d''y dy ■, , c,
^—^ + X — - — 4^^ =0, y = c^x^ -\ — -
dx^ dx^ dx X
,d^y , dy „
3. 2x' -^ + 2,x-f - 2,y = X,
dx^ dx
y — c^x + c^x f + 4^
'Xdx I
x"
- \x -i
x^Xdx.
, d'^y , dv
5. x^--^ + 4x-f + 2y = e^, y = j,;(«-2) + . . . + P^y,
In like manner, the next expression whose derivative is to be
subtracted is Q^y'-"-'''^, the next remainder being
Q^y^n-i) + p^y{.n-i> + . , , + p^y^
and so on, the values of Q^, Q^, etc., being
Q. = P.- Ql, Q, = P,- QI, etc. ... (2)
The final remainder is Q„y ; and the condition of exactness is
that this shall vanish, that is to say, Q^ = o. If this condition
be fulfilled, the integral will be
,2„J^(«-I) + Q^y^»-2) + ... + Q„_^yf + Q^_^y = Wdx + C (3)
§ XII.] EXACT LINEAR EQUATIONS. 1 37
where Q, = P„, Q, = P,- P^, Q, = P,- P/ + P:', and in
general,
Qr = Pr- Pr'-. + Pr" -. - . . . ± Po^'''^ ',
and the condition of direct integrability written at length is
(2„ = /'« - Z'^'-, + P/,. - . . . ± i^o'"' = o. . . (4)
133. For example, to determine whether the equation
is exact, we have, by the criterion, equation (4),
^3 = 4 — 14 + 16 — 6 = 0;
hence the equation is exact ; and, forming the successive values
of the coefficients Q by the equations (2), we find
which is a first integral of the given equation.
Again, on applying the criterion to this result, we obtain
^x — lOjr + 6jr = o ; hence it is also exact, and its integral is
found, by the same process, to be
{pfi — x)-^ + (2x'^' — r)y ■= — I- c^x + c^,
dx X
in which a second constant of integration is introduced.
This last result is not exact, for 2x^ — i — {yc^ — i) is not
equal to zero ; but it is a linear equation of the first order.
138 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 1 33.
and its solution gives for the complete integral of the given
equation,
xy^iipc^ — i) = see-'.* + Ci\j{x^ — i)
+ c^\og\_x + v'(«' - i)] + c^
134. The condition of direct integrability, equation (4),
Art. 132, contains the ^th derivative only of the coefficient of
the ^th derivative of y in equation (i) ; whence it is evident
that the product
dxr
is an exact derivative when s is a positive integer less thmi r.
For example, x''D^y is exact, because the fourth derivative of x^
is zero ; its integral is
x^Ifiy — jyX^D^y + dxDy — 6.
When s is negative, fractional, ox an integer equal to or
greater than r, a term of the form x^D^'y, in equation (i), gives
rise, in equation (4), to a term containing x^-'^. From this it
is evident that, if, in the given equation, we groUp together the
terms of the specified form in such a manner that s — r has
the same value for all the terms in a group, it is necessary, in
order that the equation may be exact, that each group should
separately constitute an exact derivative. If a single group be
multiplied by x'", and equation (4) be then formed, we shall
have an equation by which in may be so determined that the
group becomes exact ; but, when the given equation consists of
only one group, it becomes a homogeneous linear equation when
multiplied by xf-\ and it is more readily solved by the methods
already given for such equations.
IBS- When an equation containing more than one such
group of terms is not exact, it may happen that each group
§ XII.] INTEGRATING FACTORS OF THE FORM X"". 1 39
becomes exact when multiplied by the same power of x. For
example, the equation
2^(^+ i)g + ^(7^ + 3)£-3;' = ^. • . (i)
contains two groups of terms, in one of which j — ^ = i, and
in the other s — r =. o. Multiplying by x"', and then substi-
tuting in equation (4) of Art. 132, we have
—3*" — 7(»« + 2)jr"' + ' — 3(»z + i)x'«
+ 2{in + 3)(»2 + 2)x'"-^i + 2{ni + 2)(;« + i)a;'« = o,
which reduces to
{m + 2) {2m — i)x'« + ' + (m + 2) {2m — i)x"' = o, . (2)
the two terms in this equation respectively arising from the
two groups in equation (i). If, now, the value of in can be
so taken as to make each coefficient in equation (2) vanish,
equation (i) becomes exact when multiplied by x"'. In this
instance there are two such values of m ; namely, — 2 and J.
Using the first value of ;«, we have the exact equation
2(^ + ,)^ + ^ + 3)^ -^y = ^,
dx^ \ xjdx X? x^
whose integral is
.(.+ ,)|+^S+_3)^ = |J^.+ ..;. . . (3)
and, using the second value, we have the exact equation
2(xi + xh^ + (7^^ + 3xt)-^ - sxiy = xiX,
dx^ dx
whose integral is
2x^(x + i)^ — zx^y = x^Xdx — c^^. . . . (4)
dx
I40 LINEAR EQUATIONS : VARIABLE COEFFICIENTS. [Art. 1 35.
Having thus two first integrals of equation (i), its complete
integral is found, by elimination of y' from equations (3) and
(4), to be
x\Xdx. . (s)
5(^ + i)y — CiX + C2X ^ + *
—dx — X '
Symbolical Treatment of Exact Linear Equations.
136. The result of a direct integration is, when regarded
symbolically, equivalent to the resolution of the symbolic
operator into factors, of which that most remote from the
operand y is the simple factor D. For example, the two
successive direct integrations effected in Art. 133 show that
{xi - x)D^ + (8^== - i)D'- + \/\xD + 4
= Z'^[(x3 - x)D + 2X^ — i];
and, from Art. 135, we infer the two results,
2x'{x + i)D^ + x{^x + 3)2? — 3
= x'D\_2{x + i)D + 5 + 3^-']
= x~iD\_2xi{x + i)D — 2xi].
137. If, in a group of terms of the kind considered in
Art. 1 34, m be the least value of r, and q — mhe the constant
value oi s — r, the group may be written
x3{Aa + A,xD + A^seD" + . . .)D'<'y, . . . (i)
where A^, A^, . . . , are constant coefficients, and g may be
negative or fractional. Using &, as in Art. 123, to denote the
operator xD, the expression in parenthesis may be reduced to
the form /{&), and the group to the form
xtf{%)D'»y (2)
§ XII.] SYMBOLIC TREATMENT OF EXACT EQUATIONS. I4I
It is shown in Art. 134 that, if m is not zero, and q is zero or
a positive integer less than m, every term in the expression (i),
and hence the whole expression (2), is an exact derivative. The
symbolic transformation expressing the result, in this case, may
be effected by means of the formula deduced below.
138. We have, by differentiation,
dx dx dx^ dx'
or
B&y = ■aDy + Dy ;
whence symbolically
^D = D(& - i) (i)
Operating successively with & upon both members, we derive
^^D = QDifi - i) = D{& - I)^
ffW = &D{9 - \Y = D{& - 1)3 ;
and, in general,
f^D = D{& - ly.
Now, since /{&) consists of terms of the form A&'', it follows
that
A&)B = £>/{& - i).* (2)
* The formula by which the homogeneous linear expiression is reduced to the
form /{■9)j/ is readily deduced from this formula. For equation (i) may be written
xDy zz £>{& — i)y;
and, multiplying by x,
x'D'y = i9(i? - lb.
Changing the operand y to Dy, and using equation (2),
x^D^ = iS(i> - i)Dy - £>(& — i){-a — 2)y.
Multiplying again by x,
x^D^ = i9(iJ _ i)(i? - 2)y;
and in like manner, we prove, in general,
xrDry = ij()> _ i)(iJ — 2) , . . {S — r + l)y.
142 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art, 1 38.
Again, operating with each member of this equation upon
D (which is equivalent to changing the operand from y to Dy),
f{&)D^ = nf{& - i)£> = £>'/{& - 2).
In like manner,
/{&)I)i = I)Y{& - 2)D = Z>^/i& - 3);
and in general,
/{&)£)'" = nx'/i^ — pt) (3)
139. If ^ is a positive integer less than 7h, we can, by this
formula, write
xf/{&)I)"' = xfI)f/{& - q)D'"-^;
whence
xtf{&)D»' = &(& - i) ...{&- q -^ i)/(0- - g)D'»-9,
in which the expression for the group is reduced to the same
form as when ^ = o. We may now remove one or more of the
factors of D'^-^to the extreme left of the symbol, thus effecting
one or more, up to m — q, direct integrations, under the
condition that m is not zero, and that q has one of the values
o, I, 2 ... m — I.
The equation giving the result of m — q integrations is
x9f{p)D»' = D'"-9{& — m + q) ...{& — m + ;)/(^ — m).
140. In every other case, the possibility of resolving the
operator into factors of the required form depends upon the
presence of a proper factor in f{&). To show this, we have, by
differentiation,
Dxi^^y = xi'^^Dy + (? + ^)x^y;
whence, using Z>,r? + ' as a symbol of operation,
^^(O^ + ? + i) = Z?.V7+' (i)
§ XII.] CONDITIONS OF DIRECT INTEGRABILITY. I43
Now, if —{q -\- l) is a root of the equation f{&) =: o, so
that we can write
/W = {& + ?+ i)(&), (2)
we shall have
x'f{&)£>'" = I}xf^'(j) {&)£)'» (3)
We have thus a second condition * of direct integrability, and
an expression for the result of integration.
141. If the first member of a differential equation be
expressed in terms of the form x^f{%)D"'y, the conditions
given in Arts. 139 and 140 serve to show at once whether
the equation can be made exact by multiplication by a power
of X. For example, equation (i) of Art. 135, when written in
the form considered, is
^(2* + i)Dy + (2& + 3)(& - I);; = X.
The first term becomes exact, in accordance with the first
condition, when multiplied by x-'^ ; and the presence of the
factor {& — i) shows that the second term is also made exact
by the same factor. Hence, by equation (3), Art. 138, and
equation (i), Art. 140, the symbolic operator may be written
x^Dl{2& + 5) + x-'{2& + 3)].
* This condition might be made to include that of the preceding article ; for
we might first, by means of equation (3), Art. 138, make the transformation
xgJ\dr)Dm = xq-m x'nD'»/(d — m),
and then the expression for xfD'", in terms of i?, which is
■d{^ — 1) . . . {■& - m + I),
would, under the previous condition, contain the factor ^ -^ q — m + i, which, in
accordance with equation (i), should accompany x^ ~'"- But, since under no other
condition would this happen, and since the factor would not appear in /{9 — m)
unless 1? + ^ + I had been a factor of /(i9), this transformation is clearly un-
necessary.
144 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. I41.
Again, both terms of the last factor fulfil the condition of
Art. 140 when multiplied by x^, and the expression becomes
The value of y obtained by performing upon X the inverse
operations in the proper order is
y =
x^ + Olfl
' Xdx
■rdx.
in which each integral sign implies an arbitrary constant. The
expression is readily identified with that given in Art. 135.
It will be noticed that whenever an equation becomes exact
when multiplied by either of two different powers of x, it is
also susceptible of two successive direct integrations.
Examples XII.
Solve the following differential equations : —
I. {x — i)-^' + {x" + i)-^ + 2xy = o,
gW + x^x — i)j = r,
ei3^+x^x + C2.
e^dx + Cj,
, ,\d^y dy ,
y = c^x + y/(i — x^){C:, — sin-';c), or
y = c,x + \j{x^ — i)|^j - log[.«; + ^{x^ - i)]|.
§ XII.]
EXAMPLES.
145
dH , d'y . dy
4. — ^ + cos X — =i — 2 sin ^ ^^ — y cos x = sin 2x,
dx^ dx^ dx
y = e-
tfsin^(<:jjt: -f- C:,)dx + c^e-^'^''
sin;*; — I
5- ^zrl + (^ - 3)~ + 4^:/ + 2y = o,
dx^ dx^ dx
" %-dx
x^
' -rdx
.-^ , I -rax ,
x^ ] x^
6. ^(;c + 2)g + ^(^ + 3)|^ - iy = x,
/ X \i { xi / (X \ .
/ d'y , dy ,
.M^ = ieU^ + c.
■ 4 4(x — i)
c^ - 4,r, log
. X
'x — I J
10, Find three independent first integrals of the equation /" = X.
"-\
/'= \Xdx+c„
xy — / =
xXdx + C2:
x^y — 2xy' + 2y
x^Xdx + f,.
146 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 141.
II. Derive (a) the complete integral of/" = X from the above
first integrals, and (/3) the integral of /" = X in like manner.
(a), 2y = x^-
(13), 6y = xi
Xdx — 3.r^
Xdx — 2X
xXdx 4- 3^
xXdx +
x'Xdx + C. F.
x'iXdx + C. F.
12. Solve the equation
A^^^ A. ,^
dx^ dx
(a), as an equation of the first order for /; {ft), as an exact equation
when multiplied by. a proper power of x.
(a), y = A +
(ft), y = A +
Bx
{2\JX + 1)^
{2\lx 4- i)-
[{2slx^^2Yxdxdx.
\Jx
Bx
+
(2\lx + l)= {2slx + l)^_
13. Show that the equation
i'ljx +
sJxXdxdx.
(2^'t + 6x'i)f'' + (13*3 + 4i„-v:i)/"
+ (iijc^ + 54x2)/' — (lox — 6xi)y — 2y = X
may be written
{& + iY{2& + i)(^ - 2)y
+ x^{z» + i){2& + 3)(^ + 2)Z>)' = X,
and find its integral.
y
x^(x^ + 3)srt!x
dx
xf{x^- + 3)V
dx
2X'^
Xdx.
§ XIII.] LINEAR EQUATION OF THE SECOND ORDER. I47
XIII.
TJie Linear Equation of the Second Order.
142. No general solution of the linear differential equation
with variable coefficients exists when the order is higher than
the first : there are, however, some considerations relating
chiefly to equations of the second order which enable us to find
the integral in particular cases, and to these we now proceed.
If a particular integral of the equation
£+^l + e^ = "- «
in which P and Q are functions of x, be known, the complete
integral, not only of this equation, but of the more general
equation
S + ^l + e>-^- (')
can be found. For let j/, be the known integral of (i), and
assume
y = y\v
in equation (2). Substituting, we have, for the determination
of the new variable v,
d^v , dy, dv , d'^y^
dx^ dx dx dx^
dx dx
+ Qy,v J
= ^. . . . . (3)
The coefficient of v in this equation vanishes by virtue of the
hypothesis that j, satisfies equation (i) ; thus the equation
148 LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 42.
becomes a linear equation of the first order for -— or v'. Hence
ax
v may be determined ; and then
y = y.
t/dx + Cjj'i
is the integral of equation (2), the other constant of integration
being involved in the expression for v.
143. As an illustration, let the given equation be
(i - x")^ + x"^ — y = x{\ — x")^,
dx^ dx
in which, if the second member were zero, y ^=. x would
obviously be a particular integral. Hence, assuming y = xv,
and substituting.
or
djf_
dx
dHi , I , x^ \dv , ,-.
dotp- \ I — x^jdx
+ (- + — ^V=(i -^)i
\X I — x'/
Solving this equation, we have
•^ , f ,
(Z; = \x^dx +c,,
(l - x')^ J '
i
or
|=M.-«-)' + .i^^'.
and, integrating,
§ XIII.] A PARTICULAR INTEGRAL KNOWN. 149
Hence
y — —^x{i — x^p + c,[^xsm-'x + (i — je^)^] + c^x.
144. The simplification resulting from the substitution
jj/ = j/^v is due to the manner in which the constants enter
the value of j/ in the complete integral. For we know that 7
is of the form
y = ^i7i + '^2y2 + y,
where j, and y^ are independent particular integrals of the
equation when the second member is zero, and F is a particular
integral when the second member is X. Hence the form of v is
and that of 1/ is
V = c^ + c, )
Ji yi
-'■feJ-O'^
so that the equation determining v' must be a linear equation of
the first order. In like manner, whatever be the degree of a
linear equation, if a particular integral wh^n the second member
is zero be known, the order of the equation may be depressed
by unity.
Expression for the Complete Integral in Terms of y^.
145. The general equation for v' , where y in the equation
'ill^p^ + Qy = X (I)
dx^ ax
is put equal to y^v, and y, satisfies
^ + /'J+.<2J = 0, (2)
ax^ ax
ISO LINEAR EQUATION OF THE SECOND ORDER. [Art. I45.
is [equation (3), Art. 142]
dx
-1 + /a ^
: \j, dx
7.
Solving this linear equation of the first order, we ha,ve
, \pd:c , f \pdx „ ,
y^e^ V = jCid'J Xdx + c^ ;
and, since y = y^v = y.
v'dx,
. -\Pd:t:
€■> '^ Xdx^- + ^ij, + iTzJCi
-|P
y ax
Although of the first order, equation (i) is not so simple as
equation (2), which has the advantage of being linear. In fact,
the transformation just mentioned is advantageously employed
in the solution of an equation of the form (i). See Art. 193.
Since the complete integral of equation (2) is of the form
y = c.Xi + C:,X^ (4)
where X^ and X,, are functions of x, that of equation (i) is of
the form
_ c^X' + c^X ^l __ X' + cX:^ , .
'' c,X^ + c,X, X/+cX,' ■ • • • ^^^
which indicates the manner in which the arbitrary constant c
enters the solution.
The particular integrals of (i) produced by giving different
values to c correspond to independent integrals of equation (2),
that is to say, integrals in which the ratio c^ : c, has different
values ; the integrals in which c = o and ^ = co in the expres-
sion (s) corresponding to the integrals X, and X^ of equation (2).
77ie Transformation y = vf{x).
152. If, in Art. 142, we replace y, by w„ an arbitrary
function of x, the result is that the equation
^'y+P^ + Qy = X (I)
ax^ ax
156 LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 5 2.
is transformed, by the substitution
y = W{!>, (2)
into
where
'^ + P/JL+ Q,v = X„ (3)
ax^ ax
P. = ^^^ + F, (4)
Wi ax
& = -^5^' + ^$^ + C, (s)
z£/i ax^ Wi ax
S
X. = - (6)
/'„ (2,, and X^ are here known functions of x ; thus the equation
remains linear when a transformation of the dependent variable
of' the form j/ = v/{x) is made.
153' The arbitrary function w, can be so taken as to give to
P^ any desired value ; thus, if P^ is a given function of x, we
have, from equation (4),
^ = iCP. - P)dx;
whence
W^ =■ e^' ^ ^J (7)
Substituting in equations (5) and (6), we find, for the values of
Q, and X„ in terms of P^,
Q.=Q + ii^.' - P') + \{^ - ^) • • • (^)
and
}:\Pdx
X, = X'— (9)
§ XIII.] THE TRANSFORMATION y =. vf(^X). 157
These equations may be used in place of equations (s) and (6)
when Wi is given, P^ being first found by means of equation (4).
154. Equation (4) may be written
P^ = 2— logze/, + P\
ax
hence, when P is a rational algebraic fraction, if tw, be taken of
the form ef^""^, where ,f(x) is a rational algebraic function of x,
P, will also be a rational fraction. From this and equation (8)
it is manifest that, if the coefficients of the given differential
equation are rational algebraic functions, those of the trans-
formed equation will have the same character when a/, is of the
form e^'^''^, f(x) being an algebraic function.
In particular, if the transformation is
y = e^^'"v,
we have, since log w^ = ax'",
P, — 2max'"-^ -f P;
and then, from equation (8),
Q^ = tn^a'x^"'-^ + viax'-^-^P + m{m — i)(ur'«-^ + Q.
If, for example, this transformation be applied to the equation
'Hi _ 2bx^ -V b\xy = o,
dx^ dx
we have P = —2bx and Q = b^x^ ; whence
/", = zwajc"'-' — 2bx,
158 LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 54.
If we put 7« = 2 and a = \b, P^ vanishes, and Q^ reduces to b ;
thus the transformed equation is
;^^ + '" = °'
of which the integral is
V =■ A cos x\jb + B sin x^b.
Hence that of the given equation is
y = eVix^ (^A GO'S, xsjb + Bwxxsjb'), '
agreeing with the solution otherwise found in Art. 128.
Removal of the Term containing the First Derivative.
155. If, in Art. 153, we take P^ = o, the transformed
equation will not contain the first derivative. Distinguishing
the corresponding values of w, Q, and X by the suffix zero,
equation (7) gives
Wo = e > ; (i)
so that the transformation is
y = ve ^^ , (2)
and the transformed equation is
in which, by equations (8) and (9), Art. 153,
Qo = Q -■ iP^ - ' '^-^ (4)
2 ax
X.= xM^ (5)
§ XIII. j REDUCTION TO THE NORMAL FORM. 1 59
If the transformation y r= iv,v is followed by the similar
transformation v = w.jl, where w^ and w^ are known functions
of X, the effect is the same as that of the single transformation
y = zv^iuji, which is of the same form. It follows that the
equations which are derivable from a given equation by
transformations of the form y = vf{x) constitute a system
of equations transformable, in like manner, one into another.
Among these equations there is a single equation of the form
(3) which may thus be taken to represent the whole system.
Accordingly equation (8), Art. 153, shows that the expression
for Qo, in equation (4), has an invariable value for all the
equations of the system. The expression is therefore said to
be an invariant for the transformation y == vf{x).
156. One of the advantages of reducing an equation to the
form (3), which may be called the normal form, is that, if any
one of the equations of the system belongs to either of the
classes for which we have general solutions, the equation in
the normal form belongs to that class. For, in the first place,
if, in any equation of the system, P and Q have constant values,
equation (4) of the preceding article shows that Q^ will also be
constant. In the second place, if any one of the equations of
the system is of the homogeneous form
dx^ X dx x^
A B
putting P =. —, and S = — in equation {4), we obtain
X x^
^ _ 4,B — A^ + 2A .
4X^
hence the transformed equation is of the homogeneous form.
l6o LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 5 7.
157. As an example of reduction to the normal form, let us
take the equation
— ^ — 2 tan ;e -^ — {a^ + i)v = o.
Here jP = — 2 tan x ; therefore, by equations (i) and (4),
Art. 155,
1 tan X dx
Wo = e' = secA-,
and
Qa = — (a^ + i) — tan^jc -|- sec^a: = —a^.
Thus the transformed equation is
d^v
— — aH' = o,
dx^
The integral of this is
hence that of the given equation is
y = sec x{c^e'''' + c^e-""^.
Change of the Independent Variable.
158. If the independent variable be changed from xto z, z
being a known function of x, the formulae of transformation are
dy _ dy dz
dx dz dx '
and
d'^y _ d^yf dzV dy d^z
dz\
dx' dz^\dx) dz dx'
§ XIII.] CHANGE OF THE INDEPENDENT VARIABLE. l6l
Making these substitutions, the equation
dll^pdj.^Qy^X (I)
aoo' ax
is transformed into
\dxj dz^ ydx"- dxjdz ^-^ , . . . K^,
which is still linear, the coefficients being expressible as
functions of 2.
159. If it be possible to reduce a given equation by this
transformation to the form with constant coefficients, it is
evident, from equation (2), that we must have ( — ) equal to
the product of ^ by a constant. For example, given the
equation
(i — x^)-f- — A-^^ + ttfy = o,
dx' dx
in which Q = ; if transformation to the required form
I — jf ^
be possible, it will be the result of putting — =
dx v'(i — x") '
whence z = sin -'jr. Making the transformation, we obtain
~ + f'i'y = o,
dz^
which is of the desired form. Its integral is ,
y = A cos mz + B sin mz ;
hence that of the given equation is
y ■= A cos m sin -^x + B^va.m sin - ^x.
1 62 LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 59.
In like manner, if it be possible to reduce the equation to
the homogeneous linear form, we must have ( -^ 1 equal to
the product of Qs^ by a constant. But this transformation
succeeds only in the cases in which that considered above also
succeeds ; for it gives to log z the same value which the
preceding one gives to s ; accordingly it is equivalent to the
latter transformation followed by the transformation z = log 4,
which is that by which we pass from the form with constant
coefficients to the homogeneous form (see Aft. 123).
160. We may, if we choose, so take z as to remove the term
containing the first derivative. Equating to zero the coefficient
of this term in equation (2), Art. 158, we find, for the required
value of z,
z = e~^ ^dx.
Using this relation to express x zs> 2^. function of z, the
transformed equation is
dz^ Kdzr \dz
Examples XIII.
Solve the following differential equations
d^y , , ^dy
dx' ex
y = c^e^ +■ c^{pfi + 3jc^ -Y dx ■\- ()).
2. ^ _ ^3^ + a-(;, _ I) = o,
dx^ dx
y = c^x -\r c^x
^dx ,
e^ h I.
x^
§ XIII.J
EXAMPLES.
163
3- £-'!+<'- ■)'='.
y = c^e^ + c^e':
?2 ^''(/^ + '(f, coslog A- + ^2 sin log a:)-
164 LINEAR EQCATJON OF THE SECOND ORDER. [Art. 160.
12. x^—^ — 2nx-^ -\- (n^ 4- n + a'x^-)y = o,
y = x"(c,cosax + c^ sin ax).
13. --=^ + tan jc ^ + ycos'x = 0, y = f, sin(sin;c + r,).
ax^ ax
14. (a^ + A-^) — ^ 4- j;^ — ;«M; = o,
dx^ dx
y = c,\_x 4- VC^" + •*"')]"' + c^\x — \l{a^ + jc^)]'«.
15. JC^— 4' — 2(a^ 4- X')^ 4- (a-^ + 2X + 2)y = o,
^, (/^v , 2 c/i' , , _ sin «a: , cos «A-
«a:^ .t dx X X
17. (l — X^) — ^ — 2X^-- -I =^^ = O,
dx^ dx I — .r^
a . \ -\- X , . a . \ -\- X
y ^ c^ cos — log ^ c sin — log .
' 2 ° \ — X ^ 2 ^ \ — X
o d'y , 2 dy , a' a , . a
18. -^H ^4 y = o, y = Ci cos - 4- <^2 sin - .
19. — ^ 4- (tan A — i)^ -^ — «(« — i)jsec''A- = o,
dx^ dx
y —- ^ g{ft ■ i) t'ln X _l_ ^ ^ — « tan .r^
20. (a^ 4- a;^)^'^ 4- 2x{d' 4- a-^)— + '^'j' = o,
dx^ dx
21. Derive equation (3), Art. 147, in the form
y.^ - y^ = Ae-\"''\ from 'f^ + /'^i^ + ^^ = o,
dx dx dx^ dx
by eliminating Q and integrating the result.
§ XIII.] EXAMPLES. 165
22. Find the symbolic resolution of D' corresponding to the
integral x of the equation D'y = o.
^^ = (^ + i)(^ - ^);
23. Find the symbolic resolution of Z*^ — i corresponding to the
integrals cosh x and sinh x of the equation (Z>^ — i )j = o.
Z>^ — I = (Z> + tanh x) {D — tanh x)
= (Z) + cothit:)(Z' — cotha:).
24. Show that the ratio i' of two independent integrals of
ax^ ax
satisfies the differential equation of the third order
? - iS) - '«"
where Qa is the function defined in Art. 155.
25. Show that, if /" be expressed in terms of z, the equation of
Art. 160 may be written
H'S
+ Qy = x.
26. Prove that, in the equation
dy j_ pdy
the function
__^ + /.g + e. = o,
is an invariant with respect to the transformation z = ^{x).
1 66 SOLUTIONS IN SERIES. [Art l6l.
CHAPTER VII.
SOLUTIONS IN SERIES.
XIV.
Development of the Integral of a Differential Equation in Se?-ies.
i6i. In many cases, the only solution of a given differential
equation obtainable is in the form of a development of the
dependent variable y, in the form of an infinite series involving
powers of the independent variable x. Moreover, such a
development may be desired, even when the relation between
X and y is otherwise expressible. If we assume the series to
proceed by integral powers of x, an obvious method by which
successive tetms could generally be found is as follows. Sup-
posing the equation to be of the «th order, and assuming, for
the 71 arbitrary constants, the initial values corresponding to
X ^^ o oi y and its derivatives, up to and inclusive of the
{n — i)th, the differential equation serves to determine the
value of -^ when x = o. Differentiating the given equation,
dx"
■ • d^^^v
we have an equation containmg —, which, in like manner,
serves to determine its value when x ^^ o, and so on. Thus,
writing out the value of y in accordance with Maclaurin's
theorem, we have the values of the successive coefficients in
terms of n arbitrary constants.
§ XI V.J LINEAR EQUATIONS. 167
162. It would usually be impossible to obtain, in the manner
described above, the general term of the series. We shall
therefore consider only the case of the linear equation (and
such as can be reduced to a linear form), in which case we have
a method, now to be explained, which allows us to assume the
series in a more general form, and, at the same time, enables
us to find the law of formation of the successive coefficients.
Since we know the form of the complete integral of a linear
equation to be
y = ^ly, + <^2yz + . . . + („y„ + Y,
our .problem now is the more definite one of developing in
series the independent integrals j„ jf 2 • • ■ >"«, of the equation
v/hen the second member is zero, and the particular integral Y
of the equation when the second member is a function of x.
No arbitrary constants, it will be noticed, will now occur in the
coefficients of the required series, except the single arbitrary
constant factor in the case of each independent integral.
Developmefit of the Independent Integrals of a Linear Equation whose
Second Member is Zero.
163. We have seen, in Art. 122, that if, in the first member
of a homogeneous linear equation whose second member is zero,
we put y = Ax'", the result is an expression containing a single
power of ;r ; so that, by putting the coefficient of this power
equal to zero, we have an equation for determining m in such a
manner that y = Ax"' satisfies the differential equation, A
being an arbitrary constant.
If we make the same substitution in any linear equation
whose coefficients are rational algebraic functions of x, the
result will contain several powers of x. Let us, for the
present, suppose that it contains two powers of x, and also
1 68 SOLUTIONS IN SERIES. [Art. 1 63.
that the differential equation is of the second order. The term
containing — ^ in the differential equation will produce at least
one term, in the result of substitution, involving m in the
second degree ; hence at least one of the coefficients of
the two powers of x will be of the second degree in m. Let
X'"' and a'"'-f •>■, where s may have any value, positive or negative,
be the two powers of x, and let the coefficient of x'"' be of the
second degree. Now let m be so determined that the coefficient
of x'"' shall vanish, and suppose the quadratic equation for this
purpose to have real roots. Selecting either of the two values
of ni, the coefficient of x'"'^^ will, of course, not in general
vanish.
Suppose, now, that we put for y, in the first member of the
differential equation, the expression A^x'" + ^,;i;"' + ^ the result
will contain, in addition to the previous result, a new binomial
containing A^, and involving the powers x""-'^' and x'"'^'^^ ; the
entire coefficient of ,r"''+^ will now contain A^ and A^, and may
be made to vanish by properly determining the ratio of the
assumed constants A^ and A^- In like manner, if we assume
for y the infinite series
y = A^^ + A^x«'+^ + ^ja;'« + " + . . . ,
or
y = Sr^r.«'" + ",
we can successively cause the coefficients in the result of
substitution to vanish by properly determining the ratio of
consecutive coefficients in the assumed series. If the series
thus obtained is convergent, it defines an integral of the given
equation ; and, since in the case supposed there were two
values of m determined, we have, in general, two integrals.
If s be positive, the series will proceed by ascending powers,
and, if s be negative, by descending powers, of x.
§ XIV.] DETERMINATION OF THE COEFFICIENTS. 1 69
164. For example, let the given equation be
-^-- — x^^ — py = o (l)
The result of putting A^x'" for y in the first member is
m{fn — i)AaX'"-^ — {m + p)A^"' (2)
The first term, which is of the second degree with respect to
m, will vanish if we put
m{m — 1)^0 = o (3)
The exponent of x in this term, or m', is m — 2, and the other
exponent, or m! + s, is in ; whence j = 2. We therefore assume
the ascending series
y = l'^ArX'" + ";
and, substituting in equation (i), we have
- 2r— -^
'S,"\(m + 2r){m + 2r — i)A^x"' + '-
— {m + 2r + p)ArX>" + ^''l = o, (4)
in which r has all integral values from o to 00.
In this equation, the coefficient of each power of x must
vanish ; hence, equating to zero, the coefficient of x"' + '''-', we
have
{m + 2r){pi + 2r — i)Ar — {m + 2r — 2 + p)Ar_, = o. (5)
When r = o, this reduces to equation (3) and gives
;« = o or m = 1 ;
and when r > o, it may be written
^^^ ., + ,, 3+^ • • • W
{i/i + 2r){m + 2r — i)
which expresses the relation between any two consecutive
coefficients.
I70
SOLUTIONS IN SERIES.
[Art. 164.
When m = o, this relation becomes
A - p -V- 2r - 2 .
2r{2r — I)
whence, giving to r the successive values r, 2, 3 . . ., we have
. _ / + 4 . _ /(/ + 2)(/ + 4) .
The resulting value of y is
+ /(/+ 2)(/+ 4)^ +
(7)
Again, giving to m its other value i, the relation (6) between
consecutive coefficients becomes
whence
/ + 2r — I
(ar + i)2r
Br-^;
p + I
2.3 °'
» ^ + 3 J, (/+ !)(/ + 3) „
■O2 = — r^ -Oi = r-i -Do.
4.5
S!
H ^ + 5 z? (/+ !)(/+ 3)(/ + 5 ) „
3 - 6.7
and the resulting value of y is
7!
^ + (/ + i)^ + (/ + i)(/ + 3)^ +
3! 5 !
(8)
§ XIV.] CONVERGENCY OF THE SERIES. 171
Denoting the series in equations (7) and (8), both of which are
converging for all values of x, by y^ and j/j, the complete
integral of equation (i) is
y = Aoy^ + Boy^ (9)
165. It will be noticed that the rule which requires us to
take, for the determination of m, that term of the expression
(2) which is of the second degree in m was necessary to enable
us to obtain two independent integrals. But there is a more
important reason for the rule ; for, if we disregard it, we obtain
a divergent series. For example, in the present instance, if we
employ the other term of expression (2), Art. 164, thus obtaining
m = — / and s = —2,
the resulting series is
y = A^-P
PiP + i)
I — X-
2
^ /(/ + !)(/+ 2)(/+ 3)
2.4
The ratio of the {r + i)th to the rth term is
_ {p + 2r — 2){p + 2r — i)
X-
2r
and this expression increases without limit as r increases,
whatever be the value of x. Hence the series ultimatcly -
diverges for all values of x.
When both terms in the expression corresponding to (2) are
of the second degree in m, we can obtain two series in descend-
ing powers of x as well as two in ascending powers ; and, in
such cases, the descending series will be convergent for values
of X greater than unity, and the ascending series will be con-
vergent for values less than unity.
172 SOLUTIONS IN SERIES. [Art. 1 66.
The Particular Integral.
i66. When the second member of a linear equation is a
power of X, the method explained in the preceding articles
serves to determine the complementary function, and the
particular integral may be found by a similar process. Thus,
if the equation is
-l - x--f - py = x\
ax^ ax
the complementary function is the value of y found in Art. 164.
To obtain the particular integral, we assume for y the same
form of series as before, and the result of substitution is the
same as equation (4), Art. 164, except that the second member
is x"^ instead of zero. Equation (5) thus remains unaltered,
while, in place of equation (3), we have
m(m. — i)^o^'"~^ = s^-
This equation requires us to put
m — 2 = \, and m{m — i)Ao = i ;
whence
m = %, and A^ = 1%.
The relation (6) between consecutive coefficients now becomes
A = P+^r+l
(2^+f)(2r+f) "-"
hence
A,
Ar = 2(2/ + 4^^ + l) j^
2(2/ + 5) ^.
7^9 ^°'
_ 2(2/ + 9) . _ 2'(2p + 5) (2/ + 9) .
^^ - 11.13 ^' - 7-9-II-I3 °''
§ XIV.] BINOMIAL AND POLYNOMIAL EQUATIONS. 173
and the particular integral is
x5-^
^2(2/ + 5) , , 2^(2/ + 5) (2/ + 9) , ,
7.9 ^ 7-9-II-I3
, If the second member contained two or more terms, each
of them would give rise to a series, and the sum of these series
would constitute the particular integral.
Binomial and Polynomial Equations.
167. If we group together the terms of a linear equation
whose coefficients are rational algebraic functions of x in the
manner explained in Art. 134, we can, by multiplying by a power
of X, and employing the notation x- — = i?, put the equation in
ax
the form
/(^)j; + .^Va («■):»' + ^VsWj + •■• = o, . . (i)
in which j„ s^ . . . are all positive, or, if we choose, all negative.
The result of putting A^x"' for y in the first member is
AoMm)x"' + AoMm)x'« + '^ + AJ^{m)x""rH + . . . . (2)
Equations may be classified as binomial, trinomial, etc., accord-
ing to the number of terms they contain, when written in the
form (i), or, what is the same thing, the number of terms in
the result of substitution (2). Thus, the equation solved in
Art. 164 is a binomial equation.
In the general case, the process of solving in series is
similar to that employed in Art. 164, the form which it is neces-
sary to assume for the series being
y = I'^ArX^'^",
where s is the greatest number, integral or fractional, which is
contained a whole number of times in each of the quantities j„
s^, etc. As before, m is taken to be a root of the equation
174 SOLUTIONS IN SERIES. [Art. 1 6/.
_/■,(;«) = o, and A^ is arbitrary ; but, when the coefficient of the
general term in the complete result of substitution is equated
to zero, the relation found between the assumed coefficients
A a, A„ A^, etc., involves three or more of them, so that each
is expressed in terms of two or more of the preceding ones.
We can thus determine as many successive coefficients as we
please, but cannot usually express the general term of the
series.
We shall, in what follows, confine our attention to binomial
equations of the second order.
Finite Solutions.
i68. It sometimes happens that the series obtained as the
solution of a binomial equation terminates by reason of the
occurrence of the factor zero in the numerator of one of
the coefficients, so that we have a finite solution of the equa-
tion. For example, let the given equation be
d^y , dy y , ,
—^ + a-f- —2^ = (i)
dx^ dx x^
This is obviously a binomial equation in which j =: i ; hence,
putting
y — ^oArX'" + ';
we have
2"j[(;7z + r)(m + r — i) — 2']ArX'"+''-^
+ a{m + r)ArX"'+'^-''\ = o.
Equating to zero the coefficient of x'" + ''-'', we have
(m + r + i){m + r — 2) Ay + a{m + r — i)^^_i = o,
which, when r ^ o, gives
(m -^ \){m — 2)Aa = o; (2)
§ XIV.] FINITE SOLUTIONS. lys^
and, when r > o,
Ar = —a ■ A.^ (i^
{m + r+ i){m -\- r - 2) '^'^ ^^'
The roots of equation (2) are m = —1 and m = 2 ; taking
jn = —I, the relation (3) becomes
A^ = -« !" ~ ' . ^r-„ (4)
r{r - 3)
in which, putting r = i, and r = 2, we have
A, = -a ~' Ao,
l(-2)
^2 = —a ° A, = o.
2(-l)
All the following coefficients may now be taken equal to zero,*
* In general, when one of the coefficients vanishes, the subsequent coefficients
in the assumed series 2o Arx*" + rs must vanish ; in other words, the value of y can
contain no other terms whose exponents are of the form m -\- rs. But, in the
present case, the assumed form is y = J," Arxr- i ; and this includes the powers
Jt^, x' . . . which we know to be of possible occurrence since the other value of m
in this case is 2. Accordingly, if we continue the series, it recommences with the
term containing x^. Thus, putting r ::r 3 in equation (4), we obtain
^3 ^Z -^(2 '*2 ^ — ,
3.0 o
which is indeterminate ; then, putting r = 4, 5, etc., we have
Ai = —a — A3, A< = — a -1 A, = a^^A}, etc.
4.1 5-2 4-5
Thus, the assumed form j = So Arxr-^ really includes, in this case, the complete
integral
y - Ao(- - - W ^3^'A - -ax 4- -3-aV 4_^3^3 + . . .V
\x zj \ 4 4-5 4-S-6 /
1/6 SOLUTIONS IN SERIES. [Art. 1 68.
so that we have the finite solution *
169. For the other solution, taking m ■=■ z, the relation (3)
becomes
whence
Br = —a- — ■ — —Br-^;
{r + 2,)r
B, = -a—B^,
1.4
B. = -aj-^B, = a^l-^B..,
B, = —a-^B, = — a3.
3.6 4.5.6
Hence
BoV:, = Bax^fi - -ax + -^a^x^ ^a^x^ + ■ ■ -V
\ 4 4-5 4-S-6 /
and the complete integral is
2 — ax / 2 3 \
y = Ao—^^ + ^o^^l^i - -ax + —a-i^ - ...j.
170. Since we have, in this case, a finite integral of a linear
equation of the second order, namely,
2 — ax
y^ = >
* In like manner, if, in a trinomial equation, the coefficients between which the
relation exists are consecutive, a finite solution will occur when two consecutive
coefficients vanish.
§ XIV.] EXAMPLES. 177
equation (4), Art. 146, gives the independent integral
, 2 — ax
(2 — ax)
■dx.
We must therefore have y^ ^ Ay^ -\- By^ where y^ and y^ are
the integrals found in the preceding articles, and the constants
A and B have particular values to be determined. Since both
J// and y^ vanish when ;ir = o, while y^ does not, we shall have
A := o; and, comparing the lowest terms of the development
of the integral with the series /„ we find j5 = |- ; hence ^
2 — ax( x'e-"^ J x'r 2 , ^ , , 1 '
ax = — I ax + -^—a'x' — , . . .
X J„ (2 - axy 6L 4 4-5 J
Examples XIV.
Integrate in series the following differential equations : —
I. xp^ +ix + n)^ +{n + i)y = o,
dx^ ax
y = A(n — {n + i)x + (n + 2)— — (n + 3)^ + . . . j
+ .5.«'-«/iH ^—x +
[;'-«{ I
n — 2 (« — 2) (« — 3)
H ^ x3 + . .
(n - 2){n - 3)(« - 4)
2. -j£-^xy = o,
dx'
y = A(i- ^> + 14^ _ hAd^ + ..,)
\ 3! 6! 9! J
+ B(x-^x* + ?^xr-...y
1/8 SOLUTIONS IN SERIES. [Art. 170.
3. T.x' —^ — x-^ + (i — x')y = x^,
tljC CirX
y = Axil H + 1 h . . .)
\ 2.5 2.4.5.9 2.4.6.5.9.13 /
+ Bxi(i + ^ + -^^ + -^ + . . .'\
\ 2.3 2.4.3.7 2.4.6.3. 7. II /
1 + h • • •
1.3 1.3.3.7 1.3.5.3.7.11
4. x —^ + 2 ^ + a^x'^y = 2,
= a(\ — —a^x^ + ^a^xfi — . .\
\ 4! 7! I
\ 3! 6! ^ ;
V 5! 8! 11!
5. — ^ + ''^•s^Jl' = I + a;,
./ ax'' , rt^j;* a3^i2 \
y = A[\ ■ -I [-...)
\ 34 34-7-8 3.4. 7.8. II. 12 /
, „ / ax'' , a'x^ a3x" , \
+ Bxl 1 1 — f- • • • )
\ 4-5 4-S-8-9 4-S-8-9-I2.I3 /
x''/ _ ax* a'^x^ _ \ x^l _ ax* a^x^ _ \
2\ 5-6 5.6.9.10 ' ' '/ 6\ 6.7 6.7.10.11 ' /
6. x'^ + {x + n)^ + {n - i)y ^ x^-'',
dx^ dx
y = A(i- ^^^' ^ 4- ^-^^ - - '^^=^ ^ + ..\ + Bx^-"
\ 11 i! n + I 2 \ n + 2 ^\ J
X'' -"I I X I I x^ _ \
2 — n\ 3 — « 2 (3 — «) (4 — ;z) 3 ■ ■ ■/■
§ XIV.] EXAMPLES. l'J()
, d^y , dy ,
dx^ dx
V 3! 5! 7! 9! /
y
\ 6- 5 ' 7 ' y '
8. x-p^^ + {x + 2x')^ - 4y = o,
dx' dx
y = Axil -^-^x + -3^^. _ .4:^^3 + . . \ + 5/2 _ A + l\
\ 5 5-6 5-6-7 / V 3^ 3/
x^
Show also that a;-^(i — a;)t is an integral.
10. {^x^ — i4x^ — 2x) — - — {Sx^ — "jx + i) —
dx' dx
+ {6x — i)y = o,
y = Axi(i + 2x) + B{i — x).
11. x'^ + x'^ + (x - 2)y = o,
dx^ dx
A , „ ,/i I a; , I jc^ 1. x^ , \
X \t, 41 52! 63! /
12. Denoting the integral in Ex. 11 by. Ay^ + By^,, find, by the
method of Art. 146, an independent integral, and express the. relation
between the integrals. y, ^ ,^_ /2_ ^ ^ ^ ^\ ,^^ _ ^^_
13. x' — - — x' -^ + (x — 2)y = o,
dx' dx
V 4 4-S 4-S-6 ./ . \^ 2/
Show also that x - V-»^ is an integral.
l80 SOLUTIONS IN SERIES. [Art. I/O.
14. x^U - 4x)'p + [(i - n)x - (6 - 4«)^]^
ax^ ax
+ «(i — w)^^ = o,
/ ^(?g + 3) , , ^(« + 4)(« + 5) , , \
y = ^^» ( I + ^jc H j x^ -1 —j x^ + . . . I
+ £{1 - nx -\ —^ x^ ^-| x3 + .. .j.
y = Ax-ifi --x + —\ + Bxifi - -^x + -^—x' - . . .V
\ 5 20/ \ 1.7 1.2.7.8 J
16. (a' + x')^ + x^ - ny = o,
dx' dx
V 2 ! fl' 4 ! a* J
I n' — 1 x^ (n^ — i)(n^ — a) x* \
■^H' + -jra^ + - ii -7^ + --}
17. Denoting the integral given in Ex. 16 by Ay^ + By^, show
that |-^ _|_ ^^^2 ^ ^2)j« _ ^„y^ _^ na»-'y^,
and find the corresponding result when n = o.
log [x + sj{a- + x^)-\ = log« + ^-i^+i^-^-...
a 2 3a3 2.4 5«5
18. Expand sin (a sin-' a:) and cos (a cos-' a:) by means of the
differential equation ,3 ,
(i - ■=»^) Ji - x-j- + ^'J' = °'
«a:^ ax
of which they are independent integrals.
sm
/ a^ - I («=> _ i) («= _ o) \
(a sin- 'a;) =«.*:( I —j — x' + — j X* — ...]>
cos (a sin- 'a;) = i -x^ -i ^^ -^^x* —
2 ! 4 !
§ XV.] CASE OF EQUAL VALUES OF m. l8l
XV.
Case of Equal Values of m.
171. If the two roots of the equation determining m are
equal, we can determine one integral of the form y = ^A^x'" + ''^
by the process given in the foregoing articles ; but there is no
other integral of this form. We therefore require an independ-
ent integral of some other form.
For example, let the given equation be
x{i - x')-^^ + (i - 3x')^ - xy = o, . . . (i)
a binomial equation, in which we may take s = 2, or s = —2.
Assuming ^„ ^
we have, by substitution,
"^iim + 2ryArX»'+^^-^ — {m + 2r + iyArX»'+'''+^2 = °-
Equating to zero the coefficient of x'" + "'-^, we have
(m + zryAr — (m + 2r — i)M^_i = 0.. . . (2)
Putting r ^ o, m'Ao = o ; whence
m = o,
the two values of m being identical. Putting m = o in equation
(2), the relation between consecutive coefficients is
_ (2r- lY ^ .
whence we find the integral
Aoy. = Ao(i + -^x- + ij^^ + ^S£*' + ••■)• • (3)
^ 2^ 2^.4'' 2^.4=". 6^ /
1 82 SOLUTIONS IN SERIES. [Art. 1 72.
172. To obtain a new integral, we shall first suppose the
given equation to be so modified that one of the equal factors
in the first term of equation (2) is changed to m + 2r — h, so
that one of the values of in becomes equal to ft, while the other
value remains equal to zero. We shall then obtain the complete
integral of the modified equation, in which, after some trans-
formation, we shall put h = o, and thus obtain the complete
integral of equation (i).
The altered relation between consecutive coefficients may
be written
(m + 2r) {m + 2r — h)
in which, for a reason which will presently be explained, // is
put in the place of h. Hence, when m = o, we have
(2r — i)^
2r(2r — k')
A — \^r — 1;- .
and the first integral now is
^' = ^ + ^— ^-vV*' + —, T^TT vT^" + • • • • (5)
2(2 — h) 2.4(2 — /? )(4 — « )
Putting m ^ kin equation (4), we have
B^ = (2^ - I + hy ^ .
(2r + h){2r - h' + h)
and the second integral is
SI , (i + hy ,
y^ = x^[i -\ '^ — — — '- ■*•'
''y ' (2 +A){2 - A'+ A)'
+ (I + ^)-(3 + ^y ^ + ...Y (6)
§ XV.] CASE OF EQUAL VALUES OF m. 183
The object of introducing ,^'in equation (4), in place of the
equal quantity h, is that, when equation (6) is written in the
form
y^ = x^(}l),
^{Jt) shall be such a function of h that, by equation (5),
y, = V(o).
Developing y^ in powers of h, we have, since x^ = ^'"g*,
y^ = (i + /ilogx + . . .)[jc. + #'(0) + . . .J;
hence the complete integral is
y = Aoy, + Boy. + ^o^^C^.log^ + f (o) + ...];
or, replacing the constants A^ + B^ and Bok by A and B,
y = Ay, + By, log X + Bxf,'{o) + ..., ... (7)
in which we have retained all the terms which do not vanish
with k, and, when h = o, y, resumes the value given in
equation (3).
173. It remains to express ^'{o) in terms of x. In doing
this, we may, since k' is finally to be put equal to zero, make
this substitution in the value of \p{h) at once, and write
^^ ■' (2 + ky (2 + /iy{4 + Ay^ ^ ^
Denote the coefficient of ;ir^''in this series by 11^, so that Ho^= i,
and when r> o,
H ^ (I + hyji + hy ...{2r- 1 + hy . ,^.
(2 + hy{4 + hy ... {2r + hy ' ' ' ^^'
1 84 SOLUTIONS IN- SERIES. [Art. 1 73.
then
and
an ah.
tJTT
in which unity is taken as the lower limit because — --° = 0.
ah
' ' ' ' From equation (9),
d log Hr _ 2 2
7i ~" '. — T "r ! — 1 "I • •
+
dh X + h 3+.^ 2r — 1 -\- h
2 + k 4 + h " ' 2r + h'
which, when k = o, becomes
d log Hr
dh
_ 2 2 2 ^ _ i _ ?. .
13 2r — I 2 4 ' ' * 2r'
whence, putting h = o 'ys\ equation (10), and denoting i/''(o),
when thus expressed as a series in x, by y,
2^\l 2/ 2^.4^Vl 3 2 4/
Hence, when h =: o, equation (7) gives for the complete integral
of equation (i)*
y = Ay, + jB(y,logx + y'),
where jj/j and j/'are defined by equations (3) and (11).
* For the complete integral when we take s = —2, see Ex. XV. 7.
§ XV.] INTEGRALS OF THE LOGARITHMIC FORM. 185
Case in which the Values of m differ by a Multiple of s.
174. When the two values of m differ by a multiple of s, the
initial term of one of the series will appear as a term of
the other series ; and the coefficient of this term will contain
a zero factor in its denominator. Hence, unless a zero factor
occurs in the numerator,* the coefficient will be infinite ; and,
as in the preceding case, it is impossible to obtain two inde-
pendent integrals of the form '^A^x"' + ". For example, let the
given equation be
x^(i + x)—^ + x^ + (i — 2x)y = o. . . . (i)
dx^ dx ^ '
Putting y =z AoX"' in the first member, the result is
Ao{m- + i)x"' + Aa{ni^ — m — 2)x'" + K
Choosing the second term as that which is to vanish by the
determination of m, because the first would give imaginary
roots, we have
m — —1 or m = 2, and .f = — i ;
hence, putting y = sr^r^'"~''j
1a\{fn — r + i){m — r — 2)ArX"'-'- + ^
+ l(m — ry + i]ArX«'-''\ = o;
and, equating to zero the coefficient of x'"-''-^^,
(m — r -\- i){m — r — 2)Ar + \_{m — r + i)^+ i]^?-_i = o. (2)
* It is immaterial whether the zero factor in the numerator first occurs in the
term in question, or in a preceding term ; the result is a finite solution. An example
of this exceptional case has already occurred in Art. 168, where J = i, and the
values of m differ by an integer.
1 86 SOLUTIONS IN SERIES. [Art. 1 74,
When m^=. — i, the relation between consecutive coefficients is
r(r + 3)
and the first integral is
\ 1-4 I -2 •4-5
2.5.10 , ,
1.2.3.4.5.6
Putting m =■ 2, the relation is
and the second integral takes the form
B.y, = Box'fi S_^-. + _^ X-.
\ —2.1 — 2( — 1).1.2
__ ^.2^ ^_3 ^___
•)• (3)
■2(-l).O.I.2.3
.), (4)
in which the coefficient of ;f-' is infinite. Thus, the second
integral of the form S^^;i;'" + " fails, and we require an inde-
pendent integral of 'some other form.
175. To obtain the new integral, we proceed as in Art. 172.
Thus, supposing the second factor in the first term of equation
(2) to be changed to m — r — 2 — /i, so that the second value
of m is now 2 + /i instead of 2, and using // as in Art. 1 72, the
relation between consecutive coefficients now is
^ ^ (m-r+iY+i
§ XV.] INTEGRALS OF THE LOGARITHMIC FORM. 1 8/
When m =■ —I, this becomes
A = ^ + 1 A
and we have
2
\ 1(4 + -^)
+ ^-^ -x—-..\ (6)
Putting m =■ 2 + k, the relation between the coefficients in y^
B = (r - 3 - ^)' + I • ^ .
and the new value of B^y^ is
B.y. = B^^^4. - ^-\-'^iy ,, ^- + . . •>
in which the first term which becomes infinite when ^ = o is
"""(-Z -h)(-l-h) (-/i) (I + h'- h) (2 + h'- h) (3 + h'- h) ^"
Denoting the coefficient of this term by —, and the sum of the
preceding terms in y^ by T, we may write
Boy. = BoT
^ h \ (I - A)(4 + h'- h) ^ I ^ '
If now we write this equation in the form
Boy. = BoT + ^x^ik),
1 88 SOLUTIONS IN SERIES. [Art. 1 75.
equation (6) shows that y^ = i/'(o) ; hence the complete integral
may be written
y = A^y, + B^T -f f (i + /^log;« + . . •)[;'i + /^fCo) + . . .J.
or, putting A for the constant A^ + —.
h
y = Ay,+ B^T + By,\ogx + ^f (o) + (9)
In this equation we have retained all the terms which do not
vanish with h ; from the value of B, as defined by the expres-
sion (7), we see that, when h = o,
= \b,; .... (10)
(-2)(-i). 1.2.3 6
and, when ^ == o, we have, from equation (4),
T = 0^ + \x + \ (11)
176. The expression for i/f'(o) as a series in x, which we
shall denote by y' , is found exactly as in Art. 173. Putting
^' = o at once, in the value of \^(Ji) as defined by equation (8),
we have
^w = .-(i - (^ -;;;' + ^ x-'
r(i -hy ^. iir(2-/^)' + ii ,^_, \.
and, writing this in the form
we have H^ = i, and, when r> i,
^ r(i - ky + i]r(2 -Ay + i-\... Ur - hy + i1
^^ {I - h){2 - h) . . .{r - h){^ - h) . . .{r -{- i - h)
§ XV.] INTEGRALS OF THE LOGARITHMIC FORM. 1 89
Hence
in which
d log Hr _
dh
dn
2(1 — h) 2(2 — h)
(i — hy + 1 (2 - hy + I
(12)
2{r- h) _j_ I _|_ I ^
{r — hy + 1
h 2
+ -^+-^+...+
4 — h
f + 3 — A
When /i = o, this becomes
1 ^ ___2 4^ _
J i^ + I 2^ + 1
-Jo
dlogl/r l _
dh
r" -\- \
12 r 4 5
+
r + l
hence, putting /^ = o in equation (12), we have
y = X-''
'^(1 _ i _ l\;.-x
_I.4\2 I 4/
1.2.4.5V2 51245/
+
(13)
Now, putting /« = o in equation (9), substituting Bo = ^B
from equation (10) and the value of T from equation (11), we
have, for the complete integral of equation (i), ^
y = Ay, + B{lx^ + 3X + z + yAogx + y'),
where 7, and 7 'are defined by equations (3) and (13).
igo SOLUTIONS IN SERIES. [Art. 1 77.
Special Forms of the Particular Integral.
177. We have seen, in Art. 166, that the particular integral,
when the second member of the given equation is a power
of X, may be expressed in the form of a series similar to those
which constitute- the complementary function. Special cases
arise in which the particular integral either admits of expression
as a finite series, or can only be expressed in the logarithmic
form considered in the preceding articles. In illustration, let
us take the equation
^^ - "^>S - *l = ^""^ (^>
of which the complementary function is A sin-';tr + B. Putting
y = 'S,oA^x"'+"; we have
^Ar\^{m + 2r){m + 2r — i)x'"+^'--^
— {m + 2ryx"'+^'''] = px"; (2)
whence, when r> o,
(m + 2r)(m + 2r — i)Ay — {m + 2r — 2)'Ar_^ = o,
and the relation between consecutive coefficients is
A - (m + 2r- 2Y . , .
(;« + 2r)(m -\- 2r — i)
For the complementary function, we have m := i, or m = o.
Putting m =■ I in equation (3),
J _ (2r- ly . .
2r{2r + i)
whence
y, = x[ 1 -\ x^ -{ X* + . . . 1 . . . (4)
•^ \ 2.3 ^ 2.3.4.5 ^ J • V4y
% XV.] SPECIAL FORMS OF THE PARTICULAR INTEGRAL. I9I
This is the value of sin-'x The series corresponding \.om =.0
reduces to a single term, so that
y^ = I.
For the particular integral Y, we have, from equation (2),
whence
m = a + 2, and Ao = ^
{a + i)(« + 2)'
Putting m = a + 2 in the relation (3),
^ _ (« + 2ry .
(a + 2r + i) (« + 2r + 2)
hence
Y = —J^^l^^—(i + (^ + ^y x'
(«+i)(a + 2)\ (a + 3) (a + 4)
(« + 3)(« + 4)(« + 5)(« + 6) y ^^^
This equation gives the particular integral except when a is
a negative integer ; for instance, when a =■ o, and p =: 2, it
gives
Y = x^ii +^x' + -?^x* + ...],
\ 34 34-S-6 /
which, as will be found by comparing the finite solution of
equation (i) in the case considered, is the value of (sin-';ir)^
178. Now, in the first place, if « is a positive odd integer,
all the powers of x which occur in F occur also in j)/, ; and, when
this is the case, we can obtain a particular integral in the form
of a finite series. For example, ii a = 2> we have
y = P^(, +11^^ + ^IJL^ + .\.
4.5 V 6.7 6.7.8.9 /
192 SOLUTIONS IN SERIES. [Art. 1 78.
If we write this equation in the form
2.3/ 2.3.4.5 \ 6.7 /
the second member is equivalent to the series y^, equation (4),
with the exception of its first two terms. Thus
3Z_
2/
= y.- (x + ^^x\ or K = ^-^y. - ^(x +'^A ;
and, since the first term of this expression is included in the
complementary function, we have the particular integral
Y = X x^.
3 9
This finite particular integral would have been found directly
had we employed a series in descending powers of x.
179. In the next place, when « is a negative odd integer,
the initial term of jv, will occur in Y with an infinite coefficient.
Thus, if « = — 3 in equation (5), Art. 177, the second term
contains the first power of x and has an infinite coefficient.
To obtain the particular integral in this case, suppose first that
« = —3 + k ; then equation (5) gives
Y
px-
(-2 + /0(-i + h)
+ ;»(-! + hYx'^'^ I (i + hy
(-2+^)(-I+^)^(l+y%)V "^(2+/^) (3+/%)
Putting
Mh) = x{\ ^ ^
^^ ' \ ^(2+/^)(3+yi)
+ 'y ^-f...
§ XV.] SPECIAL FORMS OF THE PARTICULAR INTEGRAL. I93
equation (4), Art. 177, shows that 7^ = i/^(o) ; and we may write
F = r + ^(i + h\ogx + ...)[;,,+ h^'(o) + . . .J
where A^ is a quantity which remains finite when h = o.
Expanding, and rejecting the term --_y„ which is included in
h
the complementary function, we may now take, for the particular
integral,
r = r + Ny.Xogx + iVf (o) + . . . ,
in which we have retained all the terms which do not vanish
P P
with h. When h = o, the values of T and iV are — and -
2X 2
respectively ; and, finding the value of i/''(o), as in Arts. 173
and 176, we hav«>^ for the particular integral,
2X
P . , pxT Y' 1 2 I l\
+ -sm-'.«. logic + — — ( A-^
I^3^ /2 2 I I I i\ "1
+ i^TsV ^3~2~3~4~ sr' + •••]•
180. In like manner, when « is a negative even integer,
the term containing x", corresponding to y^, occurs in F with
an infinite coefficient. Thus, if « = — 4, the second term of
the series in equation (5), Art. 177, is infinite. But, putting
a = —4 + //, we have
Y = px--*h V(-2 + hY
(-3 + h){-2 + h) (-3 + h)\-2 + //)(-! + h)h
X(i ^h\ogx+ ...)(i + f -X- + . . . \
V (i +^)(2 ^-h) I
or
Y= T+ ^(i + hlogx + ..M{h).
194
SOLUTIONS IN SERIES.
[Art. 1 80.
In this case, when \\i{h) is expanded in powers of h, the first
term is unity, and there is no term containing the first power
of h ; hence, rejecting the term — which is included in the
complementary function, and then putting It. = o, we have
the particular integral
Y' - P 2/
Examples XV.
Integrate in series the following differential equations : —
I. X-^ + ^ + J)/ = o,
ax^ ax
;; = (^ + ^log^)( I - ^ + ^ - ^^^
i^ 1^.2^ 1^.2^.3^
+ 2B
"£ _ ^f I + A + ^L_/i + ^ + i^
_I^ I^.2^\ 2/ I^.2^.3^\ 2 3/
d'y , dy , ,
2. X — - + ^^ + /•»)' = o,
dx^ dx
y = {A + ^loga-)('i-^ + ^
\ 2^ 2^.4^
+ ^
2^ 2^.4^ \ 2/ 2^.4^6='\ 2 3/
~px' _ /^JC'* I
3- ■^^ + JC = o,
■^ ^ V 1-2 1-2^3 i-2^-3^4 /
1.2 I.2^3VI.2 2.3/
-^ + Bx'
X
1.2
XV.]
EXAMPLES.
19s
4. X' — ^ — (2X — l)v = O,
\ 1.4 1.2.4.5 I-2.3.4-S-6 /
+ 3^(4^' + 2JC + I)
_ 1-4 \ 4/ i-2.4-5\ 245/
- Bx-
^' ■^'^ "*" ■*'^'* "*" ^^^ "^ ^^■^ ~ '^^■^ ^ °'
2)
\ 3132133! / 3
+ ^,[4-/1 _ i + ,\ _ si^:/! _ i + , + i\ + . . 1
L3 i\3 4 / 32 !\3 S 2/ J
y = {A+ B\ogx)x{i + Y^i* + fS^*' + 7:^^4*' + • " •)
r I /i I i\ I."? /i ^ I 2 i\
+ ^ + .g^ — ( )x + —i-\- + - )x'
Li-2\i I 2/ ^ I.2^3\I 3123/
+
•3-7 (^
1-2^3 -4X1
■i e. I 2 2 I '
7. Find the integral of
x{i - x^)-~ + (i - 3>»')^ ~ xy z= o,
[equation (i), Art. 171,] when x> 1.
y= (A + B\ogx)x-Ui + —x-^ + ^^x-* + . . .\
\ 2^ 2^.4^ /
— 2Bx-
2' 1.2 2^.4^ \i. 2 3.4/
1 96
SOLUTIONS IN SERIES.
[Art. 1 80.
8. ^ + 2^ = 0,
I Aax' 4'a^x A^a^x^ \
(i + /^axi)
4«^
[Aaxi/i i\ A^a'x /i I I i\
Z — - + -) - - + - H (-- +
1-3 \i 3/ i-2.34\i ^2^34/
9. x^^ - (X' + 4A-) ^ + 4y = o,
dx^ ax
y = Ax^e'= + B(2x — x^ + x^ + xV^logx)
10. jf(i — x')—^ + (i — x^)-^ + xy ■= o,
dx^ dx
y = (A + B\ogx)(i - -x" - -^^x* - ^•3'-5 x^ - . . .\
•^ ^ ^ \ 2^ 2^4^ 2^4^6^ /
L2^ 2^4^V4 3/
+
2^4^6^V4 63 s/
■}
-• sd'y dy
II. 4^(1 - x)-^ - 4-^ - ;; = o,
y — (A -\- B\ogx)xHx ■\- ^ x + ^'-S' ^^ + 3'-5'-7' ^3 +
\ 2.6 2.4.6.8 2.4.6^8.10
- 5(32 - 8x) + 2Bx\^i^- - i _ iV
L2.6\3 2 6/
+
3'-5'
2.4.6.8'
/2_I_I^2_£_I_\
\3 2 6 5 4 8/
^ + . . .
§XV.]
EXAMPLES.
197
12. x^ — =^ + y = xi.
y = {A + B\ogx)(i - ^ + -^
\ 1.2 1.2'
^-3
•)
+ Bx- b\-—(i + a _ _f^/i + ^ +i\ + . . :
Li. 2 \^i 2/ i.2^3\i 23/
_ ^\n _ 4^' + 4!^^ _ _4!£ZL + . . A
V 1-3 i-3'-5 i-3'-S'-7 /
13. 2:c==--^ - (3^ + 2)/ H jt;
«jr ax X
y = Ax^fi -5^+-i:S_^EZ: - 5-3-^ ^^' + . . .^
\ 11! I. -I 2 ! I. -I. -3 3 ! /
+ fB - ilog.vYa-i + ^x'i\ - ^x-i
2 -1/ 2 , , 2.4 , \
X 2(1— X'' -i 3 X-' — . . . |.
105 \ 3-9 34-9-II /
14. Express the particular integral of the equation
(x - x")-^ + ^-I + 2y = 7,X',
dx^ ax
(a) in the form of an ascending series; (/3) in the form of a descending
series ; (y) as a finite expression. [See Example XIV. 9, for the
complementary function.]
(a) Y = -lU^lx^hlx^+..\.
5\ 6 6.7 /
(13) Y = —^ logx + 5J1; - - + —x-^ - -^x-'
( I + -X-' + -— *-^ +
5 \ 6 6.7
(r) i^
_ (i - xY
x^dx
(l - a:)S
198 THE HYPERGEOMETRIC SERIES. [Art. 181.
CHAPTER VIII.
THE HYPERGEOMETRIC SERIES.
XVI.
General Solution of the Binomial Equation of the Second Order.
181. The symbol F{a., p, y, z) is used to denote the series
I + ?L^2 + a(a+l)/3(^+l) „, ^ a(a+l)(a+2);3(/3+i)(/3+2) „, ^
1.7 1.2.7(7+1) 1-2.3.7(7+1) (7+2)
which is known as the hypergeometric series. Regarding the
first three elements, a, p, and 7, as constants, and the fourth as
a variable containing x, the series includes a great variety of
functions of x. In fact we shall now show that one, and generally
both, of the independent integrals of a binomial differential
equation of the second order whose second member is zero can
be expressed by means of hypergeometric series in which the
variable element is a power of x.
182. Using the notation of Art. 123,
d d^
x—- = &, whence x' — = ^(d^ — i),
dx dx'
we may, as in Art. 167 (first multiplying by a suitable power
of x), reduce the binomial equation to the form
f{&)y + x^cj>{&)y = 0, (i)
§ XVI.J BINOMIAL EQUATION OF THE SECOND ORDER. 1 99
in which f and ^ are algebraic functions, one of which will be
of a degree corresponding to the order of the equation, and the
other of the same or an inferior degree. If the equation is of
the second order, it may be written
(Q — a){9 — b)y — ^x'{& — c) {0 - d)y = o, . . (2)
in which q and s are positive or negative constants. Further-
more, the equation is readily reduced to a form in which q and s
are each equal to unity ; for, putting
we have
z = qoc^ and ■&' = z — ,
dz'
^ = qx' = -&, or & = sQ' ;
qsx^ - ^dx s
and, substituting, equation (2) becomes
183. We may, therefore, suppose the binomial equation of
the second order reduced to the standard form
{& - a)(& - l>)y - x{& - c){d- - d)y = o. . . (i)
Substituting in this equation
y = S"^r^'« + '-,
we have
1" Ar^im+r—a) {m+r—b)x'"+'-— (m+r—c) (m+r~d)x>"+''+^'] =0,
and, equating to zero the coefficient of x"' + '',
{m+r—a){m+r—i)Ar—{m + r—i—c){m + r—i—d)Ay_j = o.
200 THE HYPERGEOMETRIC SERIES. [Art. 1 83.
This gives the relation between consecutive coefficients,
.. _ {m — c + r — i) (m — d -\- r — 1) ^
(w — a + r) {m — 6 + r)
and, when ^ = o,
(m — a){m — b')Aa — o;
whence m = a ox m =■ b. Putting m =■ a, vie have for the first
integral
\ i{a - b + i)
, {a - c){a - c + ■L){a - d){a - d + 1) \ , .
^ ^.2{a~b+i){a-l, + 2) x+---j,^V
and, interchanging a and b, the second integral, is
^ i.2(^ -3 + i)(^-a + 2) -T-'-y VJ.-
Thus, putting
a — C = a
a — d = p
a — b + 1 = y
the first integral is
(4)
\ i-r i-2-y(r + 1) /
= X''F{a,/3,y,x), (5)
and the second may be written
y, = x^J^ia', p', y',x), (6)
§ XVI.] DIFFERENTIAL EQUATION OF THE SERIES. 20I
where
a. = b — C =a+l — 7
^' = b-d = ^ + I _ y I , (7)
y ^ b — (j!-f-l= 2 — y
and
b = a -\- 1 ■— y.
Differential Equation of the Hypergeometric Series.
184. If in equation (i) of the preceding article we put « = o,
and introduce a, /3, and y in place of b, c, and d by means of
equations (4), we obtain
. &{& - T_ +y)y - x{& + a) {& + p)y = o, . . . (i)
or, since ^ = ;ir — and &^ = x'— \- x—, in the ordinary no-
dx dx^ dx
tation
^(l-^)g+[r-*(i+a + /8)]g-a/3j=0, . . (2)
This'is, therefore, the differential equation of the hypergeometric
series, F{a, j3, y, x). Putting, also, a = o in the value of y^, we
have
y = AF{a,ft,y,x) + Bx^-yF{a + 1 - y, ^ + 1 - y, 2 - y, x)
for the complete integral of equation (2).
Since the complete integral of the standard form of the bi-
nomial equation of the second order, (i) Art. 183, is the product
of this complete integral by x", it follows that the general
binomial equation of the second order, equation (2), Art. 182,
is reducible to the equation of the hypergeometric series in v
and s by the transformations 3 = qx^ and y = z''v.
202 THE HYPERGEOMETRIC SERIES. [Art. 1 85.
Integral Values of y and y'.
185. When a ^^ b in equation (i), Art. 183, y =: y'r= i, and
the integrals y^ and y^ become identical, so that there is but one
integral in the form of a hypergeometric series. Again, if a
and b differ by an integer, one of the series fails by reason of
the occurrence of infinite coefficients. In this case, let a denote
the greater of the two quantities, then y is an integer greater
than unity, and y' is zero or a negative integer.
The coefficient of x"-"-, in F{!x', 0, y', x), is
(g + I - y) ■ . ■ (g -f- « - I - y) (;3 + I - y) • • . (/3 4- ^ - I - y)
(« - l) ! (2 - y) (3 - y) . . . (« - y)
This is the coefficient oix^*"-^, that is, of x^-^''-'l_'vn. y^, and is
the first which becomes infinite when y ■=. n. Now, putting
y ■= n — h,
and denoting the sum of the preceding terms of y^ (which do
not become infinite when h =■ o) by T, the complete integral
may be written
-^ V {i- + h){y + h) I ^ '
in which — is the product of B^ and the coefficient written above,
k
so that, when h ^ o, B has the finite value
p - _p (g + I - ^) • • ■ (g - I ) (^ + I - «) ■ ■ ■ (;8 - 1^ ,
(«-i)!(2-«)(3-«)...(-i) "^'
Putting
§ XVI.] INTEGRAL VALUES OF y AND y'. 203
we have, as in Arts. 172 and 175, ji = i/'(o); and, expanding in
powers of h, equation (i) becomes
y = Aoy, + B,T+ f (i + h\ogx + ...)[;'. + h^'{o) +...];
n
T>
or, putting A ior A^ + — and 7' for tj^'{o),
h
y = Ay, + BoT + ByAogx + By' + ..., . . . (4)
in which we have retained all the terms which do not vanish
with A.
To find _y' or \j/'{o), we have, from equation (3),
ah
whence, putting h =■ o, •
\j-.y\a. /3 1 yl
(5)
Finally, writing the complete integral (4) in the form
y = Ay^ + B-<], and taking the value of B^ from equation (3),
we have, for the second integral,
where y^ is the first integral x^F(a, p, y, x), T the terms which
do not become infinite in the usual expression for the second
integral, and j/' the supplementary series given in equation (s).
It is to be noticed that when y = i, 7" = o.
186. In this general solution of the case in which y is an
integer, the supplementary series y' is the same as the first
204 THE HYPERGEOMETRIC SERIES. [Art. l86.
integral _y„ except that each coefficient is multiplied by a quantity
which may be called its adjunct. The adjunct consists of the
sum of the reciprocals of the factors in the numerator diminished
by the like sum for the factors in the denominator. The first
term in_y, must be regarded as having the adjunct zero.
If _y, is a finite series, it is to be noticed that the adjunct of
each of the vanishing terms is infinite and equal to the reciprocal
of the vanishing factor. Thus the corresponding terms of the
supplementary series do not vanish, but are precisely as written
in the expression for y^, except that the' zero factors in the
numerators are omitted.
187. As an illustration, let us take the equation
{x' - x^)~^ + (^ - x") ^ - (i - 9'^)y = °>
which, when written in the form,(i). Art. 183, is
(^^ - i)y — x{%-^ — g)y = o,
so that a= I, b = —1, c = T„d= —^•, whence a = —2, /3 = 4,
y = 3. We have, therefore,
\ 1.3 1.2.3.4 /
2-10.4.5.6 V^i^s. _ _
1. 2.3^.4.5 \ 4.6 /
— 2.
in which the terms following the first three vanish. For the
other integral, employing . equation (6), Art. 185; because 7 is
an integer, we have
■q = yAogx x-'(i + — 4^.r^ + y',
-4.-3.2.3 V -i-i /
§ XVI.] IMAGINARY VALUES OF u, AND y8. 205
where the next term in the expression for T would be infinite.
The part of y' corresponding to the actual terms of y, is
L 1-3 \ 3 4 3/ I-2-3 \ 2 5 2 3/ J
and the part corresponding to the vanishing terms in equation (i)
is as therein written, with the zero omitted. Thus we have
y^ = X — X- + ^x^,
3 3
and
1 I 2 , r
■q= y^\ogx \- y ,
36.V 9
where
y' = }^x?- 47 .v3 + ^-x^\^ + \^x + '^J^x^ + . . .1
9 9 3 L 4-6 4-5-6-7 J
Imaginary I 'alues of a and /8.
188. We have assumed the roots a and b of f{&) =0 to be
real, but the roots c and d of i^(p) — o may be imaginary. In
that case a and /3 will be conjugate imaginary quantities, say
a ^= fj, -\- iv, yS = yu, — t'v.
The integrals will then take the form
y^ = ,... r + ^ii^Mf , +izfi+ii)nii±joi±jii.^ + . . 1
L i-v i-2-y(y +1) J
and
= ^^.-vfx + (^ + ; - y^[ + '^
L 1(2 - y)
+ r(/x + I - y)' + v'ir(^ + 2 - y)^ + ^^ ,. + . . .1 .
1.2(2 - y)(3 - y) J
206 THE HYPERGEOMETRIC SERIES. [Art. l88.
Again, when y is an integer, making the same substitutions
in equation (6), Art. 185, the second integral becomes
, = j^.log^ + (-i)v (y-i)!(y-2)! ^^ ,
where
Infinite Values of a and yS.
189. As explained in Art. 165, the function f{&) must be of
the second degree, but <^{9) may be of the first degree, the
equation being of the form
{& - a'){& - b)y - x{& - c)y ^ o (i)
The solution of this equation is included in the general solution
already given, for the equation is the result of making d infinite
in
Here ■, that is, - takes the place of x in the standard
a — a yS
form ; hence equation (5) Art. 183, gives the integral
^Yi I "^^ I «(« + i);3(g + i) a:' \
A 1-7/3 l.2.y(y+l) 13^^'")
for equation (2). Multiplying by the constant fi", and then
making p infinite, we have for the first integral of equation (i)
y^ ^X'^U+^X+ "(°+^\ ^ +..\
\ i-y i-2.7(r + 1) /
§ XVI.] INFINITE VALUES OF u. AND /3. 207
In like manner, for the second integral, we obtain
y^ = x^ + ^-yJpfa + I - y, P, 2 - y,^\ /3 = 00 .
190. Again, when (f)(&) is a mere constant, the equation
being reducible to the form
(& - a)(& - 6)y- xy^ o, (i)
it is the result of making both c and d infinite in
(^ _ a){& - b)y - ^ -(^ - c){& - d)y = o. (2)
(a — c) {a — a)
We have now for the first integral of equation (2)
X- (^ aP X a(a + l)^(/3 + l) x- \
a-^jS^l^ l.ya/3 1.2.7(7+1) a^;8^ " )'
Multiplying by a"^", and putting a = 00 , yS = 00 , the first in-
tegral of equation (i) is
y^ = x^li +—X H -^ — -x^ + . .. J
= x-FUp,y,^, a =00, ^=00,
and, in like manner, the second integral is
y, = x' + ^-yFU, |8, 2 - 7, ^y a = 00 , j8 = 00 .
If, in either of these cases, 7 is an integer, so that the log-
arithmic form of solution is required, the second integral is given
by equation (6), Art. 185, and is of the same form, except that
the infinite factors disappear after multiplication by 13" or a^/S",
and the reciprocals of these factors vanish from the adjunpts in
the supplementary series (5).
208 THE HYPERGEOMETRIC SERIES. [Art. I9I.
Cases in which a or /3 equals y or Unity.
191. The binomial equation of the first order may be reduced
to the form
{9 — a)y — x(& — c)y = 0, (i)
and, with the notation of the preceding articles, its solution in
series is
y^x-li +g:, + '^(°^+^)^ + ..\ (2)
This is, of course, the value of x"{i —xy-", or x''{i —x)-% which
is the integral in its ordinary finite form. The series involved
may obviously be written F{a, y, y, x), where the value of y is
arbitrary, and accordingly this value of y is one integral of the
equation
{^»-a)\{>-b)y-x{&-c){&-l>+i)y = o, . .(3)
since ^8 = 7 in equations (4), Art. 183, makes d =^ b — i. The
other integral of this equation is
^. = .^(x+-l^x+ /-^)(;-^+^) .-+...Y(4)
\ b — a ■'r I {b — a + \){b — a + 2) J
or
y:, = X^ + ^-yF{a. + I — y, I, 2 — y, x).
192. Equation (3) might have been solved by the method of
Art. 141 ; for it becomes an exact differential equation when
multiplied by;ir-*-' [see equation (i). Art. 140]. The result of
the first integration is
{& — a)y — x{0- — e)y = Cx^ ;
and in the second integration the value of j/ in equation (2) is
the complementary function, and that of j^ is the particular
§ XVI.] BINOMIAL EQUATION OF THE THIRD ORDER. 209
integral. Thus the hypergeometric series in which one of the
first two elements is equal to y reduces to the form assumed
when the equation is of the first order, and that in which one
of the first two elements is unity is of the form of the particular
integral of an equation of the first order when the second member
is a power of x.
TTie Binomial Equation of the Third Order.
193. The binomial equation of the third order may be reduced
to the form
(& -a)(&- b) (& - c)y - x(& -d){i)-- e) {& -f)y = o.
One of its three independent integrals is
\ I.5e I.2.S(6 + l)e(€ +1) /
where
a = a — d, P = a — e, y = a —/,
8 = « — 3 + 1, e = a — c + I,
and the other two are the result of interchanging a and b, and
a and c respectively.*
The notation f( "\ ^' x, ) has been employed for the series
involved in the value of j/^ above.
* When two of the roots a, 6, and c of /(i9) differ by an integer, so that one of
the quantities d or e is an integer, the powers of x which occur in one of the three
integrals will occur in another with infinite coefficients. By the process employed in
Art. 185 these infinite terms are replaced by terms involving logx and the adjuncts.
If both (5 and e are integers, the third integral contains terms which occur in each
of the others, with doubly infinite coefficients, and by a similar process these may
be replaced by terms involving (log^)* as well as logar. Similar results hold for
binomial equations of any order. See American Journal of Mathematics, vol. xi.,
pp. 49, 50, 51.
2IO THE HYPERGEOMETRIC SERIES. [Art. I94.
Development of the Solution in Descending Series.
194. When both of the functions / and <^ in the binomial
equation are of the second degree, that is, when a and /? are
finite, the integrals y^ and y^ are convergent for values of x less
than unity, and divergent when x is greater than unity. In the
latter case, convergent series are obtained by developing in
descending powers of x, or what is the same thing, ascending
powers of x-'^. Putting, in equation (i). Art. 183,
z = -, whence d' ^ z — = —&,
X dz
we have
(«•' + f) (^ + d)y - 0(^ + a) {&' + i)y = o;
hence the results are obtained by changing a, b, c, and d, in
the preceding results, to —c, —d, —a, and —b. Making these
changes in equations (4), and denoting the new values of a, ^,
and y by a^, /3„ and y„ we find
u-i = — c -\- a ^ a,
P, = —C + i = a + I — y,
y^=—c + d+i=a+i—fi;
and the integrals are
F, = z-^F{a„ p„ y„ z)
= x-^fL, a + I - 7, c. + I - ;§, iV . . . . (i)
F, = Z~'lF{a,', /?/, y/, z)
= x^fU, /3 + I - y,^ + I _ a, ^) (2)
§ XVI.] TRANSFORMATION OF THE EQUATION. 211
Transformation of the Equation of the Hypergeometric Series.
195. The equation of the hypergeometric series,
""^^ ~ ^^S + [y - ^(i + a + ^)]g - o.^y = o, . (I)
admits of transformation in a variety of ways into equations
of the same form, leading to other integrals still expressed by
means of _ hypergeometric series. One such transformation is
obviously y :=. x^-'-P^^i-a, /3+i-y, 2-y, -^3—),
J:
§ XVI.] THE TWENTY-FOUR INTEGRALS. 217
^,„ = x--i{x - x')-i-^-^FU-a, 7-a, i-a-/3+y, -i^Il^'j ,
J-a. = ^-"^U, a + I - y, a + I - /3, -V
J.,, = ^-PTt/^, ^ + I _ y, ^ + I _ „, i\
^,3 = ^^-y(^-i)v-a-3i?'/i_^, y_^, „+i_^^ iN
j,^ = x — y{x - l)y — -^F(l - a, y - a, /3 + I - a, ^y
Since the binomial equation of the second order can be
transformed into the equation of the hypergeometric series,
it follows that the binomial equation has in general twenty-four
integrals expressible by means of hypergeometric series.* But,
in the cases considered in Arts. 189 and 190, where a or yS is
infinite, we have only the integrals j/, andj/^-
* The twenty-four integrals are written above exactly as they arise in the
process indicated, except that the factor (—1 )'"'>' is dropped in the case of j/u
and yn, and ( — i)V — "-3 is dropped in the case of jis and j/io- Because j/i = yi
and /z =yi, tlie first and third integral of each group are equal, and so also
are the second and fourth, the omission of a factor in the cases mentioned above
causing no exception. It may also be shown, by comparing the developments in
powers of x, that the integrals of the first group are respectively equal to those
of the fourth group, and those of the second to those of the fifth group. But in
the third and sixth groups ^-j = (— O^jai and yio — (— 1)^>'22. Thus the twenty-four
integrals consist of six sets of equal quantities, as follows : —
2l8 THE HYPERGEOMETRIC SERIES. [Art. 202,
Solutions in Finite Form.
202. The condition that F(a., /3, y, x) may represent a finite
series is readily seen to be that one of the elements a or ;8 shall
be zero or a negative integer. But, since y^ = y^ the form of
jj/j shows that, if either y — a or y — /3 is zero * or a negative
integer, F{a., ji, y, x) may be expressed in finite algebraic form.
For example, one integral of the equation
2je(i - x)^ + [i - {2n + s)jc] -1- iny = o
is the infinite series represented by Fi^, n, ^, x). Here y — a is
a negative integer, and, using the form y^ the integral may be
written
(i - x)~"--^F{-i, J - n, I, x),
y^ = ys = yi3 = yis ,
JC2 =: j)'4 = yu - yi6,
ys = yi - yn = 719.
^6 = j/s = yn = yio,
j/io = j/ia = (—ify22 = (—ify^i-
Between any three integrals belonging to different sets there must exist a
relation of the form y^ = My^ + Ny^. These relations, in which the values of
M and N involve Gamma Functions, are equivalent to those given by Gauss in the
memoir " Determinatio Seriei Nostrae per Aequationem Differentialem Secundi
Ordinis," Werke, vol. iii. See equations [86], p. 213, and [93], p. 220. The
twenty-fonr integrals, and their separation into six sets of equal quantities, were
first given by Kummer, in a memoir " Ueber die hypergeometrische Reihe," Crelle,
vol. XV., p. 52. The order of the integrals is different from that given above, and
some errors involving factors of the form ( — 1)'' occur in the statement of the
equalities. The values of M and N are given by Kummer for the integrals
numbered by him i, 3, 5, 7, 13, and 14, corresponding to the integrals j„ j,,
J'S) J'6. y^, and _)/,o above.
* The case in which y — — o has already been considered in Art. iqx.
§ XVI.] SOLUTIONS IN FINITE FORM, 219
in which the second factor is the finite series
I -I ^^— -X = I + (2ra — \)x.
Hence the integral in question is
(i — a;)« + '
203. In like manner the integral y^ will be a finite series if
either of the quantities a+i— yory8+i — yis zero or a
negative integer ; and, since y^ = y^, the form of y^ shows that
if either i — a or i — /3 is zero or a negative integer (in other
words, if a or y8 is a positive integer), y^ may be expressed in
finite form. It will be noticed that the eight quantities,
°; /8. y — «) y — A a + I — 7, ^ + I - y, i — «, i — A
are the only values of the first two elements in the twenty-four
integrals ; hence the only cases in which they furnish finite in-
tegrals are those in which either a, p, y — a, or y — /3 is an
integer.
In the case of the general binomial equation of the second
order, the condition given in the preceding article, when
applied to both integrals, is sufficient to determine whether
finite algebraic solutions exist.*
* Finite solutions involving transcendental functions occur in certain cases
considered in the following chapter. See Arts. 209, 213, 214, and 217.
220 THE HYPERGEOMETRIC SERIES. [Art. 203.
Examples XVI.
I. Show that, in the notation of the hypergeometric series,
it + uY + {t- uY = 2tnFi^\n, -\n + \, \, j) ,
(/ + uY -{,t- uY = 2nt—^uFi-\n + 1, _i« + i,|, g^ ,
log(i + jc) = xF{\, I, 2, -^),
log^^t^ = ^xF{\, I, f, ;c=),
I — a;
e-= fU, k, I, fj= ^ + xJ'U k, 2, |^
= I + ^ + ^x^fI-l, k, 3, J j = etc., where /J = 00,
sin^ = xfU, ^, f, - -^^ , k = k'=^ca,
cosh.=ir(., .', 4,-1-,),
sin-'ar = xF{\, \, f, ^),
tan-'.* = xF{^, I, f, —x^^.
2. Show that
£^(a, A 7, ^) = ^i^(a + I, )8 + I, V + I, ^),
7?-(«, A y, X) = °^" t(y^+ y '^ ^(-^ + 2, /8 + 2, y + 2, ^), etc.
cos^ = ^(^, k', ^, ~], k = N=zo:>,
k = k' = 00 ,
dx-
§ XVI.] EXAMPLES. 221
3. Show that the equation
Ay + (B + Cx)^ + (D + Ex + Fx")^ = o
ax dx^
can be reduced to the equation of the hypergeometric series, and hence
that the complete integral is
where a and d are the roots of D + Ex + Fx'^ = o, a/S = — ,
F
independent integrals being related as y^ is to y^ in Art. 198.
'^ 4. Find the particular integral of the equation
(5- - a){d- - b)y - x{& - c){& - d)y = kxP,
and derive the integrals in Art. 183 from the result.
(,p-a){p-b)l ^ {p-a + i){p-l> + 1) J
Solve the following equations : —
y = AFi^h i I, X) + Bx-\ = ^---f + ^ .
' i. ^x
6. 2^(1 - x)^^ + x^- y = o,
dx^ dx
y^x{A + B\ogx)+B(2 + ^x^ + "-^^ + h^^ + ..\.
■>' "^ " ' \ 4 4-62 4.6.8 3 /
222 THE HYPERGEOMETRIC SERIES. [Art. 203.
7. Transform the series
, 8 , 8.10 , , 8. 10. 12 , ,
y =1 \ -\- 2-x + 3 x' + 4 X' + . . .
9 9. II 9-II.I3
by means of the theorem of Art. 197.
, .-a/ , 1-5 , I-3-5-7 , , \
y = (1 — x) ^[i-j X -\ x' +...].
■' ^ ^ \ ' 2.9 2.4. 9. II /
Solve, in finite form, the following equations : —
8. 2XU - x)'^Jl + (i - iix)^ - loy = o,
dx'^ dx
■^ (i - xY (i - xY
9. ^(i - ^)g + i(i - 2*)£ + ^-y = o,
y = A{i — x)^i — 6x) + Bxi{$ — 6x).
, -.d^v , dy ,
10. 2:«:(i — X) — - -\ '^ + 4y = o, '
dx^ dx
y = A{i - -L2X + 8^^) + Bx^{\ - a;)i
11. Solve the equation
d'^y , 2 — x^
dx^ (i — ■^ )
first transforming to the new independent variable z = 1 — x^.
y = A(i — «^)f + Bx{i — x^)^.
12. When a is a negative integer, the six integrals of 'Art. 200 are
all finite series, and therefore must, in that case, be all multiples of the
integral y^. Verify this when a = — i.
13. Show, by comparing the first two terms of the development,
that y^ = j'lj, and thence that
F{a., /3, 7, sin^ 6) = (cos^ 6Y — PF{y - a, y - ^, y, sin^ (9)
= (sec^(9)«i?'(a, y - /?, y, -tan==6l)
= (sec^ eyF{y — a, /3, y, -tan^ d).
§ XVI.] EXAMPLES. 223
14. From the expression for sin-':«: as a hypergeometric series,
derive
= €va.Qco^BF{\, i, |, sin^6l)
= tan^i^d, I, I, -tan^e).
15. The integrals of the equation
are sin nB and cos nB ; form the equation in which x = €v\B is the
independent variable, and thence derive four expressions, as in Ex. 13,
for each of these quantities.
%m.nB — n%Ya.BF{\ — \n,\ -{■ \n, \, sin' (9),
= n'&va.Bco^BF{\ -\- \n, x — \n, |, sitf^),
= «sine(cos6i)«--i^(i - \n,\ - \n, |, -\3XfB),
= «sini9(cos(9)-«-'i^(i + \n, \ + \n, f, -tan' 6);
cos«^ = F{^—\n, \n, \, sin' 6),
= cos BF{\ ->r\n,\- \n, \, sin' B),
= {co% BYF{-\n, \ - \n, \, -tan'^),
= {zo^B)-«F{\n, \ + \n, \, -tan' 6).
16. Denoting by R the expression
.(^ - r)g + (3.- x)| + .'^
show that the equation xt 1- it -\- xx—AR = o is equivalent to
dx \ ax I
x\x^ - i)^ + 3^(3^- i)S + (^9^ - ^)? + ^^'^ = °'
ax'' dt^ ax
where u = t^ ; and thence that
X + if . + 11:3!,. + ...=./, + If, + Il|!,. + . . v.
23 23.43 V 4' 4'.8' /
— Gauss, Werke, vol. iii. p. 424.
224 RICCATrS EQUATION. [Art. 204.
CHAPTER IX.
SPECIAL FORMS OF DIFFERENTIAL EQUATIONS.
XVII.
Riccati's Equation.
204. There are certain forms of differential equations which,
either for their historic interest or their importance in mathe-
matical physics, deserve special consideration. Of these we
shall consider first Riccati's equation and its transformations.
The equation
-^ -\-by^ = ex"' (i)
ax ^
was first discussed by Riccati, and attracted attention from the
fact that it was shown to be integrable in a finite form for
certain values of m. If we put - in place of x, and write a" for
the constant , the equation becomes
3>« + 1
J-Jf.y^ = a''X'", (2)
so that no generality is lost by assuming the coefficient b equal
to unity. The case in which the coefficient of x*" is negative
will be provided for by changing a^ to — a", that is, a to ia, In
the results.
§ XVII.] STANDARD LINEAR FORM. 225
205. In the form (2), Riccati's equation is the equation of
the first order connected, as in Art. 151, with the Hnear equation
of the second order,
a''x'"u = o; (3)
in other words, this last equation is the result of the substitu-
tion
_ 1 du
udx
in equation (2) ; and, denoting its complete integral by
U = /TiX. + c^X^, (4)
that of equation (2) is
c,X,+c,X, X, + cX,' ^^'
which shows the manner in which the constant of integration
enters the solution.
Standard Linear Form of the Equation.
206. The discussion of Riccati's equation is simplified by
using the linear form (3) ; moreover, the expression of the
results and transformation to other important forms is facilitated
by writing the exponent jn in the form 2j — 2.* We shall,
therefore, take
d'^u , s
a^x''i-^u = o (i)
dx^ ^ '
as the standard form of Riccati's equation from which to deter-
* This improvement of the notation was introduced by Cayley, Philosophical
Magazine, fourth series, vol. xxxvi., p. 348.
226 RICCATPS EQUATION. [Art. 206.
mine the independent integrals X^ and X^ ; the integral of the
equation in the original form being then given by equation (5)
of the preceding article.
Substituting in equation (i)
u = S"^^a;« + =«"■,
we have
Equating to zero the coefficient of ;ir'« + 2?'— ^^ we have
(jn + '^q'r) (m + 2qr — \)Ar = a'Ar-i,
and, when r = O,
m(m — i)^n = o,
whence m = o or m = i. Taking m = o, we obtain the in-
t :gral
a'' . a''
z^i = I -) :j;2? -| X'^^ + • ■ • )
2q(2q—s.) 2^.4^(2^— I) (4^— l)
and, taking m = i,
«2 = « ( I -\ -x^s + ^4? + .
V 2^(2^+1) 2^.4^(2^-1- i) (4^+ i)
207. The integrals ii^ and u^ are in no case finite series, nor
do they fulfil the condition given in Art. 202 for expression in
finite form, since in the notation there employed a and /3 are
infinite. Let us, however, apply the transformation,
u — e^'^v,
considered in Art. 154, and, if possible, determine u and m in
such a manner that the transformed equation shall still be
binomial. The equation for v is
§ XVII.] INTEGRALS IN SERIES. ■ 22/
— + 2max^—^—-\- Vm'^a'x'^"'—^ + m{m — i)ax'"—' — a^'x^i-^lv = o,
dx' dx
which, it will be noticed, becomes a binomial equation if we put
m = q and »2^a= = fl^ whence a = ± -. Hence we may put
u = e^ V,
the transformed equation then being
— + 2ax^-''—+a{q-i)xi-<'v = o; . , . . (2)
dx'^ dx
and in the results we may change the sign of a, as is indeed
evident from the form of equation (i).
208. Putting in equation (2) v = 2"^^^'""^''') we have
+ 2a{m + rq)ArX"' + ''V — i)u — fl^ — u = o J
putting m = -, and writing * and x in place of i?' and z, this
q
becomes
■S-ijy — ni)u — a'^x'^u = o, (3)
which in the ordinary notation is
d'^u m — \ dtt
dx^ X dx
— a'^u = o (4)
I X
Hence, putting q = —, and writing — in place of x^ in the
in in
six values of u given in Arts. 206 and 208, we have the follow-
ing six integrals of equation (4),
a^ X'' , a^ x^
«i = I 1 ....
m — 2 2 {m — 2) (m — 4) 2=2 !
«^=^«.^+_^£!+ ?1 ^+.
\ m -\- 2 2 {m + 2){m + 4) 2=2 !
§ XVII.] RICCATPS EQUATION.
233
u, = e^-fi - "L^^ax ^ {m-x){m- 3) g^ _ \
\ m — 1 {in — •L){ni — 2) 2 ! /
m + i {tn + i){m + 2) 2!
. '^LnJ.ax + (^^ - OC^' -3) q^ ^
m — \ {m — i) {m — 2) 2 \
V ;« + I {m + i) (m + 2) 2 I
The factor w?'" has been omitted in writing 21^, u^, and m^, but
we still have z/i = z/3 = 7^^, and z^^ = u^ = u^.
Equation (4) is integrable in finite terms when m is an odd
integer, the complete integral being A[u^] + B[u^] when m is
positive, and A[ti^] + B[u6\ when m is negative.
214. If in equation (3) we put m = 2J> + i, and make the
transformation
, ■ « = x^v,
■we have, since i9x^ V—px* V+x^» V= x^{f)- +p)V,
('? +/)('? — / — i)» — a^x'^v = o,
which in the ordinary notation is
d^v p{p 4- 1)
dx' x^
This equation is integrable in finite terms when p is an
integer.* The case in which / = 2 occurs in investigations
concerning the figure of the earth.
* See the memoir "On Riccati's Equation and its Transformations, and on some
Definite Integrals which satisfy them," by J. W. L. Glaisher, Philosophical Transac-
tions for 1881, in which the six integrals of this equation are deduced directly, and
those of the equations treated in the preceding articles are derived from them.
234 BESSEL'S EQUATION. [Art. 21$.
BesseVs Equation.
215. If, in equation (3), Art. 213, we put m = 2n and
«^ = — I, and make the transformation u = x^y, the result is
{fy^ — n''')y -\- x''y = o, (i)
or, in the ordinary notation,
«:=■— ^ + jc-f + (^2 — «=)j); = o, (2>
dx'^ ax ^
which is known as Bessel's Equation. Making the substitutions
in the values of u^ and «„ Art. 213, and denoting the corre-
sponding integrals of Bessel's equation hy y„ and_y_„, we have
y„ = x^ (I ^-— ^ + . ^ ="
n + 1 2' (n + 1) {n + 2) 2*. 2 \
y-„ = x-4i + -L_ ^ + 1 £^ + . . ;
V « — I 2^ (« — l) (ra — 2) 24.2 !
It will be noticed that either of these integrals may be
obtained from the other by changing the sign of «, which we
are at liberty to do by virtue of the form of the differential
equation.
216. The integrals corresponding to the other four values of
u in Art. 213 are imaginary in form. Making the substitutions
in the value of tt^, we may write, since u^ = t{^ = x"y„,
y„ = i" (cos jf + i sin x) (/>„ — iQ„) ,
in which
P„= I — (2« + l)(2n+ 7,) x^
(2« + l)(2«+ 2) 2 ! '
Q^ = i!L+^x - (2^ + i) (2n + 3) (2n + s) x^
2n + 1 (2« + l)(272 + 2)(2;? -I- 3) 3 !
§ XVIL] FINITE SOLUTIONS. 235
The value of y„ derived from ti(, is the same thing with the
sign of i changed ; hence we infer that
y„ = a:" (/■„ cos X + Qn sin x) ,
and also that
Pn sin X — Q„ cos x = o*
Changing the sign of n, the other integral of Bessel's equa-
tion may, in like manner, be written in the form
where
y-„ = x-''(P-„ cos* + Q-„ sinj;),
p _ . (2n- i)(2n — s) x^
(2;z — i) (2« — 2) 2 I
Q_„ = ^"~ ^ x — (^^— i)(g" - 3)(2^ - 5) ^
zn — I (2« — i) (2« — 2) (2« — 3) 3 !
Finite Solutions.
217. The case in which Bessel's equation admits of finite
solution is that in which n is one-half of an odd integer. Taking
n to be positive, the series P-„ and (2_« contain, in this case,
terms whose coefficients have zero factors in the numerators.
Denoting by [/"_„] and [Q-„] the finite series preceding these
terms, we have, as explained in Art. 212, an integral \_y-„] in
finite form, but differing in value from j/-„. Thus
* The resulting value of tan x may be written
_ »z + 5 x^ {in -t- 7) {m -f- 9) £5 ^
m + 2 3 ! {m -I- 2) (?« -(- 4) 5 1
tan ;«: = '
_ »z + 3 ^ , (?» -I- S) ('" -^ 7) :^ _
OT + 2 2 ! (m -1- 2) (;« -F 4) 4 !
in which »« may have any value.
236 SESSEL'S EQUATION. [Art. 21/.
= x-"\co?,x [/'-bJ +smx\_Q-„']\
+ ix-'^lsmx^F-n] — C05 x\_Q-„'\\,
in which the coefficient of i does «<7^ vanish, as it does in Art. 216.
If we substitute this expression in the differential equation, it
is evident that the real and imaginary parts of the result must
separately vanish, so that we have the two real integrals
171 = a:-«^cosx[P-„J + smx\_Q-„']\,
and
■q^= X~"\'i\nx \_P-„]—C05x\_Q-,^\.
The complete integral may therefore, in this case, be written
, . 7 = Cx-"\\_P-n\ cos {x + a) + [<2-«J sin {x + a) ^,
where C and a are the constants of integration.
218. Comparing the integrals -q^. and ly^ with j/_„ and j^„.
Art. 215, it is evident that, since cosx[P-^ + sinjr[g_„J is an
even function, and smx[P-„] — cos;r[2_„] is an odd function,
the development of i/i contains only the powers of x which occur
in j;/_„, and -q^ only those which occur in y„. Moreover, the first
coefficient in -q^ is unity. It follows that -q^ =y-m and that
1/2 is the product oiy„ by a constant.*
* To find this constant, we notice that the part of the series P_ „ — tQ^„, which
is rejected from the value of ^_„, when we use the finite expressions, as in Art. 217,
commences with the term containing x'^". Denoting the coefficient of this term by
A, the rejected part of;>'_„ is Ay„. Thus
jV- „ = [y-„'] + Ay„ = rji + if,^ + Ay„.
But we have shown that tj, =y_„; hence tj, = -y^, where A is the coefficient
of jr^" in P_„ — iQ-„, that is, in — iQ—„, Art. 216, since 2» is an odd integer.
Thus
(2» — l)! (2»)!
§ XVII.] THE BESSELIAN FUNCTION. 237
The Besselian Function.
219. If, when n is a positive integer, we multiply yn, Art.
215, by the constant , the resulting integral of Bessel's
2"« !
equation is known as the Besselian function of the «th order.
and is denoted by^„. Thus
T X" f I X^ , I X4
2"n \\ « + I 2= (ra + i) (« + 2) 242 !
° {n + r)\r[\2j
More generally, for all values of n we may write
I x^ , I X*
Jn =
2«V{n + l) V « + I 2= ' (« + l) (« + 2) 242 !
= 3" {-'^y /'^Y+"''
'r(« ■\--L-\-r)r\\2J
and then, in general, the complete integral of Bessel's equation
is
where y_ „ is the same function of — n that J„ is of n. It is
to be noticed, however, that the factor which converts the
series y-n to J-n is zero in value when « is a positive integer.
Substituting the values of r)^ and y„. Arts. 217 and 215, we have, for the devel-
opment of the odd function sin .»;[/>_„] — cos.r[5_„],
2'''-'r(«-^)!r ,.«f. £i_+ ^ ...V
(2K-0!(2«)! V 2(2K+2) 2.4(2»+2)(2» + 4) /
238 BESSEVS EQUATION. [Art. 219.
The series in this case contains infinite terms which are thus
rendered finite, while the finite terms preceding that which
contains x^" axe made to vanish. The result is that, when n is
an integer,
and the expression AJ„ + 5/_„ fails to represent the complete
integral.* The second integral in this case takes the logarithmic
form, and is found in Art. 221.
220. The expression ior y„, given in Art. 216, shows that
2'T(« + i)
where
/> = I _ ^" + 3 ^ r (2« +5)(2i^ +7) X^ _
2« + 2 2 ! (2« + 2) (2« + 4) 4 !
Q^=X— ^" + 5 ^ + (2W+ 7) (2« +9 ) ^ _
, 2« + 2 3 ! (2;? + 2) (2« + 4) 5 !
* Finite expressions iox J„ and _/_„ exist when n is of the form j* + J, / being
an integer. These are multiples of tj^ and n^, Art. 217, respectively. Substituting
in the numerical factors the values of the corresponding Gamma functions, which are
r(j> + 1) = (i> + i) T{p + j) = %t'j\ V^.
and, taking account also, in the case of y^^j, of the factor found in the preceding
foot-note, we find
2p-^p\^TT X^ + h
) (2;>)! cosj:[/Lc^ + .i] + sir
2>-^/! Vir " " x^ + i
and
§ XVII.] BESSELIAN FUNCTION OF THE SECOND KIND. 239
The Besselian Function of the Second Kind.
221. The second integral, when n is an integer, may be
found by the process employed in Arts. 175 and 176, and in
similar cases. Thus, changing equation (i), Art. 215, to
('9 — «) ('9 + « — h')y + x'^y = o,
and putting y ^^S^A^x'"*'^^, the relation between consecutive
coefficients is
A,=
(m + 2r — n) {m + 2r -\- n — h')
where h' is put for h. Making m = n and m =—n + k suc-
cessively, we have the integrals
„/ x^ , X*
2{2n + 2 — h') 2.4(2^ + 2 — A')(2« + 4 — ^')
and
y^„ = x~" + ^
X'
(2 + A - h') {2n-2-h)
(^2 + h — h') . . . {2n + h — h') {2n — 2 - k) . . . {— h)
^y {2n + 2+h-h'){2+h)'^ "
Denoting the product of ^r" and the series last written by VW,
we have i/'(o) =y„, and the complete integral 7 = Aof„ + Bof-x
may be written
y = Aoyn + £cT +^{1 + hlogx + . ..)[y„ + h^'{o) +...J,
240
BESSEL'S EQUATION.
[Art. 221.
where, when h = o,
Bo
Bo
2.4 .. . 2n{zn — 2) (2n — 4) ... 2 2=«-'«! (i^ — i) !
and T denotes the aggregate of terms in y^„, which remain
finite when h = 0. We have therefore
y = Ay„ + BoT+ By„ log jc + B^'{o) + . . . ,
and may take as the second integral, when h = o,
y„\ogx— 2"'-^n\(n — i)! 7"+ i/f'(o).
If this expression be divided by 2"n !, the first term becomes
^«log;ir; denoting the quotient by Y„, and developing xj/'ip) as
in Art. 173, we have
V„=/„logx — 2''-^{n — i) \x-"
i+-
I x^
X*
x"
.!.(«+ l)
(«— l)(«— 2) 24.2!
I+^_^^
■ I x'"-^ 1
{n—i)\ 2=«-=(« — i) ij
!+ 1/2^
I \x*
i.2.(« + i)(ra + 2) V ' 2 ' n+i ' n+2j2*
1+'-+ '
+ ...
and the complete integral of Bessel's equation, when n is an
integer may be written
y = A/„ + BV„.
§ XVII.] LEGENDRE'S EQUATION. 24 1
The integral Y„ is called the Besselian function of the second
kind.*
Legendre's Equation.
222. The equation
(t. — x''\—^ — 2x-^ ■\- n(n-{-\)y = 0., . . . . (i)
dx^ dx
or, as it may be written,
; I (l — a:=)^ t + «(i^ 4- l)_y = O,
d_
dx
is known as Legendres Equation, because, when n is an integer,
it is the differential equation satisfied by the «th member of a
set of rational integral functions of x known as the Legendrean
Coefficients.! Particular interest, therefore, attaches to the
case in which « is a positive integer; and it is to be noticed
* The properties of the Besselian functions are discussed in Lommel's " Studien
iiber die Bessel'schen Functionen," Leipzig, 1868; Todhunter's "Treatise on La-
place's Functions, Lame's Functions, and Bessel's Functions," London, 1875, etc.
t The Legendrean Coefficient of the Kth order is the coefficient of a." in the
expansion in ascending powers of a of the expression
F=
y/{l — 2iu; + a2)
and is denoted by Pn{_x), or simply by /«• It is readily shown that
dx^ dx > da} da >
whence, substituting V= 'S.'i^ a»Pn and equating to zero the coefficient of a", we find
- i(i - X') ^\ + n{n + i)P„ = o.
: ( dx >
d_
dx\
When x= \, V= — ^- — = l + o + a^ + . . . ; hence Pnif) = I for all values of «.
I — o
242 LEGENDRE'S EQUATION: [Art. 222,
that this includes the case in which n is a negative integer; for,
if in that case we put — n = n! + i, whence — (;? + i) ?= «', we
shall have an equation of the same form in which n! is zero or a
positive integer.
223. When written in the i9-form, Legendre's equation is
*(* — i)j— ^H* — «) ('^ + « + 1)7 = o,
a binomial equation in which both terms are of the second
degree in »9. Hence the equation may be solved in series pro-
ceeding either by ascending or descending powers of x. Putting
y = 'S,"ArX"' + ^'', we have, for the integrals in ascending series,
y^ = I — n{n + i)— + n{n — 2) (« + i) (« + 3)'^ — .-..,
2 ! 4!
and
y^=x — {n—i){n + 2)— + (« — i) (;? — 3) (« + 2) (« + 4)^ — . . . .
Again, writing the equation in the form
(.? — n){f^ + n+ i)y — x-^f}{» — i)y= o,
and putting y = 'S,'^ArX'^~'"', we have the integrals in descend-
ing series
n(n — I )
J>'3 = ^"(i-
2(211
and
.y-, I n{n - i) (^ - 2) (n - 3) ^^,^ _ \
i) 2.4(2«— i)(2ra-3) ■ ■ 7'
_j_ (« + i) (^z + 2) (;z -t- 3) {n -f- 4) ^_^ ^ N
2-4(2«-f 3)(2« + 5) ' "/
XVII.] THE LEGENDREAN COEFFICIENTS. 243
The Legendrean Coefficients.
224. When « is a positive integer, y^ ox y^ is a finite series
according as n is even or odd ; and in either case y^ is a finite
expression, differing fromj/i orj^^ only by a constant factor. If
j/3 be multiplied by the constant
(211- i){2n- 3) ■ ■ . I Qj. {2n)\ ^
n\ 2»(«!)°'
the resulting integral is the Legendrean coefficient of the «th
order, which is denoted by f „ By the cancellation of common
factors in the numerators and denominators of the coefficients,
the successive values of P„ may be written as follows : —
5 , 3
2 2
/> =x, P, = -x^ - f *,
T-^ '^■'K ^-i
P. = —X* - 2—^= + — ,
* 4.2 4.2 4.2'
/> = —XS - 2—^3 + ^x,
5 4.2 4.2 4.2
11.9.7 9.7.5 7-5-3 ^ 5-3-I
in which the law of formation of the coefficients is obvious.*
* The constant is so taken that the definition of F„ given above agrees with
that given in the preceding foot-note. For, putting x = I, and forming the differ-
ences of the successive fractions which in the expressions last written are multiplied
by the binomial coefficients, it is readily shown that F„(j) = i, for all values of n.
244 LEGENDRE'S EQUATION. [Art. 225.
The Second Integral when n is an Integer.
225. When n is an integer, the second integral of Legendre's
equation admits of expression in a finite form.
Assume
y= uF„ — V, (i)
where ti and v are functions of x. By substitution in equation
(i), Art. 222, we have
~^du dP„
dx dx
«| {i-x^)'^^-2x'^ + n{n + i)F„\ + 2{i-x^)-
1 D ( / ^sd'^u du\ , ,\d^v , dv , , \
. +P„\ (l —X'')- 2X— \ — (l —X^)—~+2X- «(« + l)»=0,
(. dx^ dx ) dx'^ dx
in which the coefficient of u vanishes, because P„ is an integral,
and that of P„ will vanish if u be so taken that
/ ^\d'''u du
(l — X^) 2X — =: o.
dx'^ dx
This condition is satisfied if we take (i — x'') — = i, whence
dx
« = ilog^±I; (2)
X — 1
the equation then becomes
/ ^\d''v dv , , , s dP., , N
0— =);^--^ + -(« + 0^'=^^, . . .(3)
and we shall have a solution of Legendre's equation in the
assumed form (i), if v is determined as a particular integral of
this equation.
§ XVII.] THE SECOND INTEGRAL. 245
Now, since Pn is a rational and integral algebraic function
of the «th degree, the second member of equation (3) is an
algebraic function of the (fi — i)th degree ; hence the particular
integral required is the sum of those of several equations of the
form
(i — ^=)— ^ — 'zx-^-\- nin -\- \)y ■= axt., . . . (4)
dx^ dx
in which / is a positive integer less than n. Solving equation
(4) in descending series, the particular integral is
Y= '^ ("i + /(/-i)
(/ — «)(/ + « + i)V {p-\- n— \){p— n — 2)
+ /(/-i)(/-2)(/-3) ^-4 _,. _
(/ + « — i) (/ + « — 3) (/ — « — 2) (/ - ra — 4)
which, when / is an integer, is a finite series containing no
negative powers of x. Thus the particular integral of equation
{4) is an algebraic function of x of the /th degree, and that of
equation (3) is an algebraic function of the {n — i)th degree.
Denoting this function by R„, we have therefore an integral
of Legendre's equation of the form
G. = i^.log^^-i?« (5)
226. Since
1 , .X + I I I I ,
ilog— :i^ =- + —+-— + .. .,
X — \ X 2X3 ^xS
the product iF„ log , when developed in descending series,
commences with the term containing x"-^; and as R„ contains
no terms of higher degree, the development of Q„ cannot con-
tain X". It follows that, putting Q„ = Ay^ + By^ where y^ and
y^ are the integrals in descending series. Art. 223, we must
246 LEGENDRE'S EQUATION. [Art. 226.
have Qn = By^.* But j/4 commences with the term j;"""~' ; we
therefore infer that in the product above mentioned the terms
with positive exponents are the same as those of i?«, and are
cancelled thereby in the development of Q^, while the terms
with negative exponents vanish until we reach the term Bx~'^~'^.
The formation of the required terms of this product affords a
ready method of calculating i?„.f
* To determine the value of B, we notice that equation (3), Art. 147, gives, for
the relation between the integrals Pn and Qn of Legendre's equation,
p dQ„ „ dP„ _ A , .
Pn — y« — — — , (,I>
ax ax I — x'^
where ^ is a definite constant. Substituting from equation (5), this gives
Pn' +{x^-l) \p/4^ - rJ^'X = A.
L ax ax J
Putting x=l, we have A=J, because /'«(!)= I, and Pn and Rk being rational
integral functions, the quantity in brackets does not become infinite. Now, from Art.
224, P„ = ^ "'' j'3 ; substituting this value, and putting A = \, Qn = Byi, equation
2"(k !)^
(i) becomes B(2n)\ l dy^ _ ^\ i_
2»(k!)A «'■«; ^'■dx] \-x^
Developing both members in descending powers, and comparing the first terms, we
— ;^ — <—x — ' (— K — I — ») = — jr— 3,
whence d_ 2"(«!)'-'
~ (2«+ l)!'
that is Q„ = fM)Ly,
(2« + l) !
■f The Legendrean coefficients are sometimes called zonal harmonics, the term
spherical harmonics (in French and German ire&tXses fonctions spheriques and Kugel-
functionen) being applied to a more general class of functions which include them.
The function Qn is the zonal harmonic of the second kind. Discussions of the
properties of the functions Pn and Qn will be found in Todhunter's Treatise " On
Laplace's Functions, Lame's Functions, and Bessel's Functions," London, 1875 ;
Ferrers' "Spherical Harmonics," London, 1877; Heine's "Handbuch der Kugel-
functionen," Berlin, 1878 ; etc.
§ XVII.] EXAMPLES. 247
Examples XVII.
Solve the following differential equations : —
dy a^ (x — a)e''^~^+ i:(x + a)e-'"^~^
1. — 1- _j)2 = — , y = 2 i ^^ ~
d^u _8
2. -; a^x ^u = o,
dx''
u = Axe--^""''^ {i + zax-"^) + Bxe^^^^^'ii - sax'').
^_2^_^,^^o u^Ae^^ii- ax) +£6-0^(1+ ax).
dx^ X dx
4. — + - — -a=?y
^. -r- — <^ y = — '
dx^ x"
y = Ax-''e^^(^i — ax + ia^x") + Bx-''e-'"'{\ + ax + ^a'x").
d^y , , 6y
y = Ca:-=[(3 — n''x^) cos {nx + a) + yix sin («a: + a)].
248
RICCATPS EQUATION, ETC.
[Art. 226.
„ d'^y , dy , , , .
dx'^ dx ^ ^'-^ '
Ae'' -\-Be-^
six
^ d^'y , dy , ax'' —
II. x''—^ + x^ +-
dx^ dx 4^2
-JV = 0,
/~tf ^
^ Sajf? ^. 2^ \8ax?J ' ' 'J
14. Show that for all values of n
I 4- ,-y ■(- ^ + ^ ^' I (» + 2)(n + 4) x3
n -\- I 2\ (n + i){n + 2) ^\
~ 1-X + " "'"^ - - (" + ^)('^ + 4) ^ ,
« + i2! (n+ i){n+ 2) ;i\
15. Show that the complete integral of the equation
dx^ dx x^
may be written in the form
xy = A(2 — qx) + Be-f^{2 + qx).
§ XVII.]
EXAMPLES.
249
16. If in Riccati's equation a= = — i, show that the integral may
be expressed in BesseHan functions.
u = i^X
MrT'^Hir
1 7. Reduce to Bessel's form the equation
x'^ + nx-^ + (6 + cx''"')y = o,
dx^ dx ^ '•' '
and show that its integral in Besselian functions is
where ^=V(£(^^i)lZL4£]
^/4^v)+^/-.K
d'v
18. -y^+ye"'' = ny,
d^y , y
19. —^ +-^ = o,
^ dx'^ 4x
d'^y , dy ,
^f^y , dy , ,
J = x^lAJ^{x^) + BY^{x^)'\.
y = AJo{2x^) +BYj2xi).
. cos x^ , _ sm x^
y = A h B
22. Putting « = e"'/'-^'*''^) = XPrhr, show that
^.^2 x'^ dh^
and thence that 7^ +1 is an integral of
d^v , /(/ + i) „
dx^ x'^
250 EXAMPLES. [Art. 226,
23. Pm and P„ being Legendrean coefficients, show that
n{n + i) ^ P„P,ndx = J' (i - *')^ ^'^*'
and thence that
PnPmdx = O,
except when m = n. Also show that, when m + n is an even number,
P„Pmdx = o, unless m ■= n.
§ XVIIL] SIMULTANEOUS EQUATIONS. 251
CHAPTER X.
EQUATIONS INVOLVING MORE THAN TWO VARIABLES.
XVIIL
Determinate Systems of the First Order.
227. A system of n simultaneous equations between « + t
variables and their differentials is a determinate system of the
first order, because it serves to determine the ratios of the
« + I differentials ; so that, one of the variables being taken
as independent, the others vary in a determinate manner, and
may therefore be regarded as functions of the single indepen-
dent variable.
A determinate system involving the variables x, y, z, . . .
may be written in the symmetrical form
dx dy dz
in which X, Y, Z, . . . may be any functions of the variables.
228. When the system is put in this form, we may consider
the several equations each of which involves two of the differen-
tials ; if one of these contains only the corresponding variables,
it is an ordinary differential equation between two variables, and
its integration gives us a relation between these two variables.
This integral may be used to eliminate one of these variables
from one of the other equations, and may thus enable us to
obtain another equation containing only two variables ; and
252 SIMULTANEOUS EQUATIONS. [Art.> 228.
finally, in this manner, n integral equations between the n -\- \
variables. Given, for example, the system
dx _dy _dz ^ ^
y X z^
in which the equation involving dx and dy is independent of z ;
integrating it, we have
x'^ — y^ = a (2)
Employing this to eliminate x, the equation involving dy and dz
becomes
dy _dz
and the integral of this is
y + \/{y' + a) = iz (3)
The integral equations (2) and (3) containing two constants of
integration constitute the complete solution of the given system.
Transformation of Variables.
229. A system of differential equations given in the sym-
metrical form is readily transformed so that a new variable
replaces one of the given variables. For example, when there
are three variables x, y, and z, let it be desired to replace x by
a new variable u, a given function of x, y, and z. We have
dx _ dy _ dz _ \dx + ft,dy + vdz . >
where X, /u,, and v denote any arbitrary multipliers. Now, u
being a given function of x, y, z,
, du , , du J , du ,
du = — dx H dy -\ dz.
dx dy dz
§ XVIII.] TRANSFORMATION OF VARIABLES. 253
Hence, if X, ^x., v be taken equal to the partial derivatives of u,
the numerator of the last fraction in equation (i) is du, and
denoting the denominator by U, we have
dy _dz _du , ,
y~z~ £/' ^^'
in which Y, Z, and U are to be expressed in terms of y, z, and
?/ by the elimination of x.
As an illustration, in the example of the preceding article
we may write
dx _ dy _ dz _ dx + dy _
y X ' z y -\- X '
so that, taking u = x +j/, we have for one of the equations
dz _ du
z u '
of which the integral is
u = bz,
which is equivalent to equation (3) of the preceding article.
Exact Equations.
230. If X, /It, V in equation (i). Art. 229, be so taken that
\X -\- y.Y + vZ = o,
we shall have
\dx + \i-dy + vdz = o.
An equation derived in this manner may be exact, and thus lead
directly to an integral equation containing all three of the
variables.
254 SIMULTANEOUS EQUATIONS. [Art. 230.
For example, if the given equations are
dx dy dz , .
= — ,=-, ' (0
mz — ny nx — iz ly — mx
we thus obtain
Idx + mdy + ndz = 0, (2)
and also
xdx -\- ydy + zdz = o (3)
Each of these is an exact equation, and their integration gives
Ix -^ my -\- nz ■= a, (4)
and
^^+r + 2'' = b, (5)
which constitute the complete solution of the given equations.
The Integrals of a System.
231. Denoting an exact equation derived as in the preceding
article from the system
dx dy dz , .
X = Y = ^ (^)
by du = o, the multipliers A, /a, v are the partial derivatives of
the function u, and the relation connecting them is
v-du ydu ydu _ , ^
dx dy dz '
Hence, if a function u satisfies this condition, the exact equation
du = ois derivable from the system (i), and its integral
u =^ a
may be taken as one of the two equations which constitute the
solution.
§ XVIII.] THE INTEGRALS OF A SYSTEM. 255
*
An equation of this form containing but one constant of
integration is called an integral of the system in contradistinc-
tion from an integral equation which, like equation (3), Art. 228,
contains more than one arbitrary constant.
Conversely, if ;/ = a is an integral of the system (i), the
function ;/ must satisfy equation (2) : for let us transform the
system as in Art. 229 ; then, because dii = o, we shall have
U = O, which is equation (2).
232. When there are more than three variables, we can
derive in the same way a similar, condition which must be satis-
fied by the partial derivatives of the function n, when u = a is
an integral. Thus it is possible to verify a single integral of
a system without having a complete solution. The complete
solution of a system involving 7/ + i variables may be put in
the form of a system of n integrals corresponding to the n
arbitrary constants. The number of integrals is, however, in
any case unlimited ; for in the complete solution we may replace
any constant by any function of the several constants. Thus,
let
i( = a and v = b
be two independent integrals of a system involving three varia-
bles, and let ^ denote any function, then
{u,v) = (l>{a,b) = C
is a relation between x, y, z and the arbitrary constant C, and is
therefore an integral. This is, in fact, the general expression
for the integrals of the system of which n = a and v = b are
two independent integrals. Accordingly, it will be found that,
if u and v are functions of x and y satisfying equation (2) of the
preceding article, <^ {u, v) also satisfies that equation, being
an arbitrary function.
2S6 SIMULTANEOUS EQUATIONS. [Art. 233.
«
Equations of Higher Order equivalent to Determinate Systems
of the First Order.
233. An equation of the second order may be regarded as
equivalent to two equations of the first order between x, y and
/, one of which is that which defines /, namely,
dx
and the other is the result of writing ^ in place of — ^ in the
dx dx""
given equation. For example, the system equivalent to the:
equation
which is solved in Art. jQ, is, when written in the symmetrical
form of Art. 227,
/ y
I
in which the equation involving dp apd dy is independent of x,.
and thus directly integrable.
The integrals of the equivalent system are the same as the
first integrals of the equation of the second order, cf which two,
corresponding to the constants of integration employed, may be
regarded as independent. Compare Art. 79. The complete
integral of the equation of the second order, containing as it
does both constants of integration, is an integral equation, but.
not an integral, being the result of eliminating the variable p
either before or after a second integration. Compare Art. 82.
In like manner, an equation of the ^^th order is equivalent to
a system of n equations of the first order, between n + i varia-
bles. Again, two simultaneous equations of the second order
§ XVIII.] GEOMETRICAL INTERPRETATION. 257
between three variables are equivalent to a system of four
equations of the first order between five variables, and so on.
Geometrical Meaning of a System involving Three Variables.
234. Let X, y and z be regarded as the rectangular coor-
dinates in space of a moving point ; then, since the system of
differential equations
dx _dy _dz
determines the ratios of dx, dy and dz, it determines at every
instant the direction in which the point {x, y, z), subject to the
differential equations, is moving. Starting, then, from any
initial point A, the moving point will describe a definite line,
and any two equations between x, y and z, representing two
surfaces of which this line is the intersection, will form a parti-
cular solution. If we take a point not on the line thus deter-
mined for a new initial point, we shall determine another line in
space representing another particular solution. The two equa-
tions forming the complete solution must contain two arbitrary
constants, so that it may be possible to give any initial position
to {x, y, z). The entire system of lines representing particular
solutions is therefore a doubly infinite system of lines, no two
of which can intersect, assuming X, V and Z to be one-valued
functions, because at each position there is but one direction in
which the point (x, y, z) can move. We hence infer also that
the constants will appear only in the first degree.
235. Consider, now, the complete solution as given by two
integral equations between x, y, z and the constants a and b.
The surfaces represented determine by their intersection a par-
ticular line of the system. Let the constant b pass through all
possible values, while a remains fixed ; then at least one of the
surfaces moves, and the intersection describes a surface. The
258 SIMULTANEOUS EQUATIONS. [Art. 235.
equation of this surface is the integral corresponding to the con-
stant a ; for it is the result of eliminating b from the two equa-
tions, and is thus a relation between x, y, z and a. Hence, an
integral represents a surface passing through a singly infinite
system of lines selected from the doubly infinite system, and of
course not intersecting any of the other lines of the system.*
If a and b both vary but in such a manner that C = 4> {a, b)
remains constant, the intersection of the two surfaces describes
the surface whose equation is the integral corresponding to the
constant C. Compare Art. 232.
236. Thus, in the example given in Art. 230, the integral (4)
represents a plane perpendicular to the line
f = ^ = ^-, (X)
I m n
and the integral (5) represents a sphere whose centre is at the
origin. The intersection of the plane and sphere corresponding
to particular values of the constants is a circle having its centre
upon, and its plane perpendicular to, the fixed line (i).
Hence the doubly infinite system of lines represented by the
differential equations (i). Art. 230, consists of the circles which
have this line for axis ; and the integrals of the differential
system represent all surfaces of revolution having the same line
for axis.
Examples XVHI.
Solve the following systems of simultaneous equations : —
dx dy dz „ , „ 1 i ^ - 1 v
I. — = ^ = , y^ + z^ = a, logbx = tan 1..
X z y z
* On the other hand, of the surface represented by an integral equation, we can
only say that it passes through a particular Une of the system.
§ XVIIL]
EXAMPLES.
259
dx , 2X
at t
t , a
3 t^
X ■\- y ■= be'.
dx dy
dz
y + z z + X X + y
\l{x+y + z) =
z — y X —
dx __ dy _ dz
x" —y^ — z' 2xy 2XZ
y = az, X' +y' + z^ = .
_ Idx _ mdy _ ndz I'^x + my + n'z = a,
mn{y — z) nl{z — x) lm(x — y') ' I'^x'^ + tri^y" -\- n'z = b.
adx _ bdy _ cdz ax'' + by'' + cz^ = A,
{b — c)yz {c — d)zx {a — b)xy a'x' + by + c'z' = B.
dx dy
X y z — a^{x''+ y -^ z?^^
y = ax, ^'-^ = /S[3 + y'(^^ +_)»= + 2=)].
8. Show that the general integral of
dx dy dz
I m n
represents cylindrical surfaces, and that the general integral of
dx _ dy _ dz
X — a. y — ^ Z — y
represents conical surfaces.
26o SIMULTANEOUS EQUATIONS. [Art. 237.
XIX.
Simultaneous Linear Equations.
237. We have seen that the complete solution of a system
of simultaneous equations of the first order between « + i
variables consists of n relations between the n -\-\ variables and
n constants of integration. Selecting any two variables, the
elimination of the remaining n — 1 variables gives a rela-
tion between these two variables, involving in general the
n constants.
We may also, selecting one of the two variables as inde-
pendent, perform the elimination before the integration, the
result being the equation of the «th order,* of which the equa-
tion just mentioned is the complete integral.
For example, in the case of three variables, x, y and t, if we
require the differential equation connecting x with the inde-
pendent variable t, the two given equations are to be regarded
as connecting with t the four quantities x, y, — and -^.
Taking their derivatives with respect to /, we have four equa-
djC dv d^x d^v
tions containing x, y, -r-, ~, —^ and —^ ; and from these
at at af dt^
dv dv^
four we can eliminate y, ^ and -~, thus obtaining an equa-
tion of the second order, in which x is the dependent, and t the
independent variable.
238. As a method of solution the process is particularly
applicable to linear equations with constant coefficients, since
* The differential equation connecting two of the variables may be of a lower
order, in which case the integral relation will contain fewer than n constants. For
example, one of the equations of the first order may cnntain only two variables, as in
Art. 228, and then the integral relation will contain but one constant.
§ XIX.] LINEAR SYSTEMS. 261
in that case we have a direct method of solving the resulting
equations.
For example, the equations
and
J+5*+J' = ^' (i)
^--x^7,y = e''t (2)
at
are linear equations with constant coefficients, if t be taken as
the independent variable. Differentiating the first equation,
we have
d^x dx dy_ ^ .
and since -^ does not occur in this it is unnecessary to differ-
entiate the second. Eliminating ~- and y by means of equa-
cit
tions (2) and (i), we have
^ + 8^+16^ = 4^^-^=^
dt^ dt
The complementary function is {A +Bt)e-^, and the par-
ticular integral is found by the methods of section X. The
resulting value of x is
x=(^A+ Bt) e-^t + ^e* - ^e^',
and, substituting • this value in equation (i), we find without
further integration,
y = - {A + B + Bi) e- ^* + i^e^' + ^he*.
262
SIMULTANEOUS EQUATIONS.
[Art. 239.
239. The differentiation and elimination required in the
process illustrated above are more expeditiously performed by
the symbolic method. For, since the differentiation is indi-
cated by symbolic multiplication by D, the equations may be
treated as ordinary algebraic equations. Moreover, the process
is the same if one or both the equations are of an order higher
than the first.
For example, the system
d^y dx
'ir^--dt-^y = '''
dx , dy
when written symbolically, is
{2j> — 4)j)/ —Dx= 2t,
2Dy + (4Z) — i)x — o.
Eliminating x, we have, in the determinant notation,
2t —D-
o 4-O-3
(/5-i)=(2Z' + 3)j=2-f/,
2I)' — 4 —D
2D 4^-3
y =
or
and integrating.
jC = (^ + Bt) e* + Ce-lt _ ^/.
The value of x is, in this example, most readily derived from
that of y by first elimiriating Dx from the given equations, thus
obtaining
(8^?= + 2Z> — x(i)y — 2,x= 8/,
whence, substituting the value of y,
X = e\(,B —2A— 2Bf) — \ge-it- \.
§ XIX.] NUMBER OF ARBITRARY CONSTANTS. 263
240. Ordinarily, in finding the value of the variable first,
eliminated it is necessary to perform an integration, and, when
this is done, the new constants of integration are not arbitrary,
but must be determined so as to satisfy the given equations.
Thus, if in the preceding example the value of x had been de-
rived from the first of the given equations, after substituting the
value of y, it would have contained an unknown constant in
place of the terni — \, and it would have been necessary to
substitute in the second equation to determine the value of this
constant.
The value of x may also be derived directly from the result
of eliminating y, namely.
2D-^ — 4
-D
X =
2l> — 6,
2t
2D
aD-z
2D
The complementary functions for the two variables will then
be of the same form, and will involve two sets of constants.
By substituting in one of the given equations, we shall have an
identity in which, equating to zero the coefficients of the several
terms of the complementary function, the relations between the
constants may be determined.
241. The number of constants of integration which enter
the solution is that which indicates the order of the resultant-
equation. This number is not necessarily the sum of the in-
dices of the orders of the given equations, although it cannot
exceed this sum ; it depends upon the form of the given equa-
tions, being, as the process shows, the index of the degree in D
of the determinant of the first members. ,
Denoting this number by m, the values of the n dependent
variables contain n sets of m constants, of which one set is
arbitrary. Substituting the values in one of the given equa-
tions, we have an identity giving m relations between the con-
stants ; it is therefore necessary to substitute in « — i of the
given equations to obtain the relations between the constants.
264
SIMULTANEOUS EQUATIONS.
[Art. 242.
Introduction of a New Variable.
242. The solution of a system of differential equations is
sometimes facilitated by the introduction of a new variable, in
terms of which we then seek to express each of the original
variables. Given, for example, the system
dx _dy _dz
(I)
where
X=ax + by + cz + d, Y= a'x + b'y + c'z + d!,
Z=a"x + b"y + c"z + d".
If we introduce a new variable t by assuming dt equal to the
common value of the members of equation (i), we shall have
the system
dx dy dz
X
Z
dt.
(2)
involving four variables, which is linear if t be taken as the
independent variable. Writing the equations symbolically, the
system is
(a — D)x -^ by -\- cz + d =0,
a'x + (b' — D)y + c'z + d' = o,
a"x .+ b"y + (c" -D)z + d"=o;
whence
a-B b c
a' b'-D c'
a'l b" c" - D
d b
d' b'-D
c" — D
(3)
(4)
§ XIX.]
INTRODUCTION OF A NEW VARIABLE.
265
in which D may be omitted in the second member because it
contains no variable. Denoting the roots of the cubic
a-D
a'
a"
':
b
c
-D
c'
b"
c"
—
D
(5)
by \i, X2 and X3, equation (4) and the similar equations for y
and z give
y = A'^^' + B'^^* + C"/=' + ^' !- , .... (6)
z = A"e'''-* + B^e''^* + C"^'* + k"
in which k, k\ k" are the values of x, y, z respectively, which
make X = o, F= o and Z = o.
Substituting these values in the first of equations (3), we
have one of the three equations determining k, k' and k", and
for the constants of integration the three relations,
{a - \^A + M' + cA'' = o,
(a - \^B + bB' + cB" = o,
In like manner, substitution in each of the other equations
gives three relations between the constants, making in all nine
relations, of which six are independent. The three relations
between A, A' and A" are
{a ~\^)A + bA' + cA" = o,
a'A+{b'-\,)A' + c'A"=o,
a" A + b"A' + (c" - X,)A" = o,
266 SIMULTANEOUS EQUATIONS. [Art. 242.
which are equivalent to two equations for the ratios A -.A^ : A",
since their determinant vanishes because A.i is a root of equa-
tion (5).
243. The introduction of a new variable, as in the preceding
article, introduces a new constant of integration into the system,
but this constant is so connected with the new variable that the
relations between the original variables obtained by eliminating
the new variable are independent also of this constant. Thus
in the value of x, equation (6), we might have put ^ + a in place
of t, employing only two other constants ; then the relations
between x, y and z, which we should obtain by eliminating t,
would obviously contain only the two constants last mentioned.
Examples XIX.
Solve the following systems of linear equations : —
■ " ' — ay = £', -^ — X -Y dy ■= e^',
'■ ^ + 5- -. . ,^
y = iAe-'i' - Be-i* + i^e* + -^e'K
dx dy ,.
2. ■ = — i^ = dt.,
IX— y x+y
x={A+ Bi)e\ y= (A-B + B/)e''.
3. (5/ + gz)dx + dy + dz = o, (^y + 2z)dx + zdy — dz = o,
y = Ae-^+Be-7'', z= — ^Ae~^ + Be-^^.
dx ,^ dy
4. = di=-^,
— my mx
x = A cos mt ■\- B sin mt, y = A sin mf — B cos mf.
§ XIX.]
EXAMPLES.
267
5 . a h n^y = e^, -^ + az = o,
dx dx
y = Ae^'' + Be-"^ +
az= — nAe'" + nBe-"'' —
e'
«2— I
n d^x , , d^'y , .
6. 1- m^y = o, -^ — m^x = o,
df ^ dt^
mx mx
777/ ^ /«« , . ■ mx\ , y/af A mx , . ■ mx\
x = e^ I A^ cos - — f- ^2 sm -— ) + 1? ^ I A, cos- |-^4Siii— - ,
y = e^ lA^ sin- ^2C0s^- )+<• *' (^4003-^ ^3 sm
,mx\ ^/2
,-J+. p...^^ „3..„^^
dx , dy , , .
4-t: + 9 ^ + 44X + 49y = ^,
dt dt
x = Ae-' + Be-('i + ift-^- i^e*,
dt dt dt
xA A- B A — xB
yz=Acost-\-Bs\Vi.t, x= — ^ — ^ — cos^H --^— sin/.
dt^ dt dt^ dt
x = Ai cos at — A^ sin at + B^ cosftt — ^2 sin^t,
y = A^ cos at -^ Aj sin at + i9j cos /8/ + ^i sin j8/ ;
where a and 5 stand for — n ± \J {n^ -\- m'') .
268
SIMULTANEOUS EQUATIONS. [Art. 243.
dt
dx,,
dt
dx-^
dt
x^ = Ae^^* + B^* + C,
Xj + ^2 A.2 + ^2 «2
a^azA ^^t I .
a^a^B
^-*+^,
where A.i and X2 are the roots of
A* + {oi + a^ + a^)k + a^a^ + a2a3 + a^a-i = o.
dx dy
13. <
x = «6'(^ COS / + -5 sin «■) ,
;( = tf"[(^ -5) COS/ + (^ +.5) sin/].
dx , dy , , ,
— + 2-^ + x+7y=e'-3,
dt dt
-^ — 2X + T,y = 12 — 2,et,
dt
X = Ae-''* COS / + Be-'-* sin/ + fl tf' — ff ,
j);= - [A + B)e-'.t Q.o%t + {A — B)e-'-* imt — -f^et -i^ ^.
dx
13. / — ■ + 2a; — 2JC
dt
= t, ^^ + x + 5y = t\
x= At-'^+ £i-^+^t + ^t%
y=-At-^~\Bt-^-i^t + ^t-.
§ XIX.] EXAMPLES. 269
d^x d^ V
X = {A, + B^t)et + {A^ -^BJ)e-t,
d'^x dy o o. dx , dy
^ dt' dt ' dt dt ^-^ '
x={A+ Bt)e^i + Ce-^* - t,
y = (3^ -2A- 2B{)e^t - \Ce-^t - \.
16. Show that the integrals of the system
^ = ax + by + c, ^ = aJx + ^> + c\
dt dt
are
(a + m^a') {x + ^z^ji') + ^
where nii and w^ are the roots of
a'vf^ + {a — b') m — b = o ;
and obtain a similar solution for the system
^ = ax + l>y, ^ = a'x + b'y,
dt^ dt^
x+m,y = A,e^'' + '"^'''^^'+B,e-^'' + '"^'''^\
x + m^y = A,/'' + '"'''">^'+B,e-'^'' + "'^">\
2/0 EQUATIONS INVOLVING THREE VARIABLES. [Art. 244.
XX.
Single Differential Equations involving more than Two Variables.
"Zitif. When the number of differential equations connecting
« + I variables is less than n, it is of course impossible to estab-
lish n integral relations between the variables. We shall here
consider only the case of a single equation, at first supposing
the number of variables to be three; and we shall find that there
does not always exist an equivalent single integral relation be-
tween the variables.
We have seen that when there are two differential relations
between x, y and z, the integrable equations which separately
furnish the two independent relations between the variables
are generally produced by the combination of the given equa-
tions. We have now to find the condition under which a single
given equation is thus integrable, and the meaning of an equa-
tion in which the condition is not fulfilled.
The Condition of Integrability.
245. The given equation will be of the form
Pdx + Qdy + Rdz = 0, (i)
in ^yhich P, Q and R may be any functions of x, y and z.
If there be an integral relation between x, y, z and an arbitrary
constant a to which this equation is equivalent, let it be put
in the form
u =■ a,
so that a shall disappear by differentiation ; then the differential
equation du = o, or
du J , du J , du ,
— dx -\ dy -\ dz = o,
dx dy dz
§ XX.] THE CONDITION OF INTEGRABILITY. 271
must be equivalent to equation (i). In other words, if the
equation is integrable, there must exist a function of x, y and z
whose partial derivatives are proportional to P, Q and R ;
thus
dii T, du ^ du r>
ax dy dz
•NT • d du d du . ., ^- ■
Mow, since —-—- = ----, etc., these equations give
dy dx dx dy
^(dF_dQ\^Qd^_pdi,
dy dx) dx dy''
JdQ _d^^ j^djL _ QdiA.
\dz dy) dy dz
\dx dz J dz
djx
dx
Multiplying the first of these equations by R, the second by P
and the third by Q, and adding the results, i>. is eliminated, and
we have
''(f-f)^{x,y) — c.
In fact, the condition of integrability, Art. 245, reduces in
this case to
dS _dT^
dy dx
which is the same as the condition of exactness for the differ-
ential expression Sdx + Tdy. See Art. 25.
249. The most obvious application of this principle is to the
case in which one variable can be entirely separated from the
other two. Thus the example in Art. 246 might have been
solved in this way ; for, dividing by zy, which separates the
variable z, it becomes
y dx — xdy dz _
y" z
§ XX.] HOMOGENEOUS EQUATIONS. 2/5
an exact equation of which the integral is
X
log z = c.
y
Homogeneous Equations.
250. In the case of a homogeneous equation between x,
y and s, one variable can be separated from the other two by
means of a transformation of the same form as that employed
in the corresponding case with two variables, Art. 20. For,
putting
X ^ ZU, J = zv,
the homogeneous equation may be written in the form
2»<^(«, v)dx + z"\l/(u, v)dy + z"x(^) v)dz =■ o;
and, substituting
dx = zdu + udz, dy = zdv + vdz,
we have
z<^{u, v)du -f zi/f {u, v)dv + [x(w, v) + u<^{u, v) + v\p{u, »)] dz = o.
If the coefficient of dz vanishes, we have an equation between
the two variables u and v. If not, the equation takes the form
dz <^{u,v)du + \li{u,v)dv _
z ■)(^{u,v) + u^{u,v) +vxp{u,v)~
and, in accordance with Art. 248, the second term will be an
exact differential if the given equation is integrable.
251. As an example, let us take the equation
{y +yz + !i^)dx + {z^ 4- zx +■ x^)dy + (x" + xy +y^)dz = o, . (i)
2/6 EQUATIONS CONTAINING [Art. 251.
which will be found to satisfy the condition of integrability.
Making the substitutions, and reducing, we have
dz (?'= -\- V -\- i)du + (u' + u + i)dv _
z {u + v + x) {uv + u + v)
Knowing the second term to be an exact differential, we in-
tegrate it at once with respect to ti, and obtain
loff 2 — log \- C = o,
^ ^UV + U + V
The symmetry of this equation shows that C is a constant and
not a function of v : thus the integral of equation (i) is
xy + yz + zx = c(x + y + z).
Equations containing more than Three Variables.
252. In order that an equation of the form
Pdx + Qdy + Rdz + Tdt = o
involving four variables may be integrable, it must obviously be
integrable when any one of the four variables is made constant.
Thus, regarding z, x and y successively as constants, equation
(2), Art. 245, gives the three conditions of integrability,
\dy dx) \dt dyj^^\dx dt)
„(dQ dR\ f.(dR dT\ (dT dQ\ ^
^fdR _ dP\ ^ji(dP _ dT\^pfdJi _ ^^ = o.
\dx dz j \dt dx J \dz dt J
§ XX.] MORE THAN THREE VARIABLES. 277
Again regarding / as constant, we have the condition
p(dQ. _dR\ ^fciR _ dP\ .pfdF _ dQ
\dz dy) ^\dx dz) \dy
dx ',
but this is not an independent condition, for it may be deduced
by multiplying the preceding equations by R, P and Q respec-
tively, and adding the results.
253- In general, if the equation contains n variables, the
number of conditions of the above form which we can write is
ft \7Z ^— I) (^i ^— 2)
— ^^ '-S i, which is the number of ways we can select
1.2.3
three out of the n variables. But, in writing the independent
conditions, we may confine our attention to those in which a
selected variable occurs, for any condition not containing this
variable may be obtained exactly as in the preceding article
from three of those which do contain it. Thus the number
/m t\ f^f 'y\ ,
of independent conditions is ^^ '—^ —, which is the num-
1.2
ber of ways we can select two out of the n — 1 remaining
variables.
254. When the conditions of integrability are satisfied, the
integral is found, as in the case of three variables, by first
integrating as if all the variables except two were constant,
:he quantity C introduced by this integration being a function
of those variables which were taken as constants. To determine
this function the total differential of the result is compared with
the given equation. The result either determines the value of
dC in terms of these last variables (in which case dC should be
an exact differential), or else is such that the first two variables
may be eliminated simultaneously, as in the example of Art.
247, giving an integrable equation between C and the remaining
variables.
278 EQUATIONS INVOLVING THREE VARIABLES. [Art. 255.
The Non-Integrable Equation.
255. In an equation of the form
Pdx + Qdy + Rdz = o
the variables x, y and z may have any simultaneous values
whatever ; but, for each set of values, the equation imposes a
restriction upon the relative rates of variation of the variables,
that is, upon the ratios of dx, dy and dz. When the condition
expressed by equation (2), Art. 245, is satisfied, there exists an
integral equation which, for each of the sets of values of x, y
and z which satisfy it, imposes the same restriction upon their
relative rates of variation. At the same time the presence of
an arbitrary constant makes the integral sufficiently general to
be satisfied by any simultaneous values of x, y and z.
But, when the condition of integrability is not satisfied, there
is no such integral equation. Two integral equations will, how-
ever, constitute a particular solution, when, for each set of
simultaneous values of x, y and z which satisfy them, the ratios
which they determine for dx, dy and dz satisfy, in connection
with these values, the given differential equation.
256. If one of the two integral equations is assumed in
advance, the determination of the particular solutions consistent
with the assumed equation is effected by solving a pair of
simultaneous differential equations, namely, the given equation
and the result of differentiating the assumed relation. Geo-
metrically the problem is that of determining the lines upon a
certain surface which satisfy the given differential equation.
For example, given the equation
(\ -\- 2a)xdx -\- y(\ — x)dy -\- zdi == o .... (i)
(which it will be found does not satisfy the condition of inte-
§ XX.] THE NON-INTEGRABLE EQUATION. 279
grability) ; let it be required to find the lines on the surface of
the sphere
^2 _|.j,2 ^_ 22 = ^2 (2)
such that a point moving along any one of them satisfies equa-
tion (i). Differentiating equation (2), we have
xdx -\- ydy -\- zdz =^ o, (3)
which with equation (i) forms a system of which equation (2)
is one integral and a second integral is required. Subtracting,
we have an equation free from z, namely, ,
2axdx — xydy = o,
the integral of which is
y^ = 4ax + C. , (4)
Hence the required lines are those whose projections upon the
plane of xj/ are the parabolas represented by equation (4).
257. In order to form a general solution of a non-integrable
equation, the assumed equation must contain an arbitrary func-
tion. We might, for example, assume
y=A^), (i)
where / is arbitrary, because any particular solution consisting
of two relations between x, y and z might be put in the form
y=f(x), z = ^{x). ,If, therefore, we determine all the particu-
lar solutions consistent with equation (i), the result will, when
/ is regarded as arbitrary, include all the particular solutions.
The equation which completes the solution will, as in the pre-
ceding example, be found by integration, and will therefore
contain an arbitrary constant C, to which a special value !r!'j5:t
28o EQUATIONS INVOLVING THREE VARIABLES. [Art. 257.
be given (as well as a special form to the function/) in order to
produce a given particular solution.
258. The general solution of the equation
Pdx + Qdy + Rdz =0 (i)
may be presented in quite a different form, which is due to
Monge, depending upon a special mode of assuming the equa-
tion containing the arbitrary function.
Let |u, be an integrating factor of the equation
* Pdx + Qdy = o
when z is regarded as a constant, and let V= C be the corre-
sponding integral, so that
dV= ^Fdx + fiQdy.
Then, in the first place, the pair of equations
z= c, and F= C, (2)
where c and C are arbitrary constants, constitutes a class of
particular solutions of (i). Now, for the general solution, let us
assume
^='^W (3)
Differentiating, we have
t,Fdx + ,iQdy+ ^ _ <^'(z)"| ^2 = o, . . . .(4)
which, combined with equation (i), gives
' 0'(^) — iJ-Rjdz = o. . . , . . , (5)
§ XX.] MONGE'S SOLUTION. 28 1
Hence, if F= <^(^') be taken as one of the relations between the
variables, we must have, in order to satisfy equation (i), either
dz = o, or else
^ - 4><{z) -y.R = o (6)
The first supposition gives z = c and V= (c), a system of
solutions of the form (2) ; the second constitutes, in connection
with equation (3), Mongers solution.
It is to be noticed that when it is possible to determine <^ so
that equation (6) is identically satisfied, the given equation is
integrable, and V= {z) is its integral. But, in the non-inte-
grable case, <^ is to be regarded as arbitrary.
Monge's solution includes all solutions excepting those of
the form (2). To show this, it is only necessary to notice that,
with this exception, any particular solution can be expressed in
the form x =f^{z), y ^/^{z) ; and, substituting these values in
the expression for F as a function of x, y and z, we have an
equation of the form V={z) determining the form of <^ for
the particular solution in question. The particular solution is
therefore among those determined by one of the two methods
of satisfying equation (s) ; and, as it is not of the form (2), it
must be that determined by equations (3) and (6).
The distinction between this solution and that given in Art.
257 is further explained in Art. 262 from the geometrical point
of view.
Geometrical Meaning of a Single Differential Equation between
Three Variables.
259. Regarding x, y and z as the rectangular coordinates of
a variable point, as in Art. 234, the single equation
Pdx + Qdy + Hdz = o (i)
282 EQUATIONS INVOLVING THREE VARIABLES. [Art. 259.
expresses that the point (ar, y, z) is moving in some direction, of
which the direction-cosines /, m, n, which are proportional to
dx, dy and dz, satisfy the condition
/■/+ Qm +Iin = o (2)
Consider also a point satisfying the simultaneous equations
dx _
dy
dz
(i\
P
Q
R' ' ' •
' • \.6)
and therefore
moving
in the
direction whose
direction-
cosines
satisfy
X _
P
Q
_ V
. . . .
■ • (4)
Suppose the moving points which satisfy equations (i) and (3)
respectively to be passing through the same fixed point A ; then
P, Q and R have the same values for each, and equations (2)
and (4) give
A, + mfi. + nv = o,
which is the condition expressing that the directions in question
are at right angles. We have seen, in Art. 234, that equations
(3) represent a system of lines, there being one line of the
system passing through any given point. Hence equation (i)
simply restricts a point to move in such a manner that it every-
where cuts orthogonally the system of lines represented by
equations (3), which we may call the auxiliary system.
260. Now, suppose in the first place that equation (i) is
integrable. The integral represents a system of surfaces one
of which passes through the given point A. This surface con-
tains all the possible paths of the moving point which pass
through A, and every line in space representing a particular
solution lies in some one of the surfaces belonging to the system.
§ XX.] GEOMETRICAL INTERPRETATION. 283
The restriction imposed by equation (i) is in this case completely
expressed by a single equation.
Every member of the system of surfaces represerited by the
integral cuts the auxiliary system of lines orthogonally, so that
equation (2), Art. 245, considered with reference to the system
of lines represented by equations (3), expresses the condition
that the system shall admit of a system of orthogonally cutting
surfaces.
261. On the other hand, when the condition of integrability
is not satisfied, the possible paths of the moving point which
pass through A do not lie in any one surface, the auxiliary
system of lines, in this case, not admitting of orthogonally cut-
ting surfaces.*
When, as in the example of Art. 256, the point subject to
equation (i) is in addition restricted to a given surface, the
auxiliary lines not piercing this surface orthogonally, there is in
general at each point but one direction on the surface in which
* The distinction between the two cases may be further elucidated thus : Select
from the doubly infinite system of auxiliary lines those which pierce a given plane in
any closed curve, thus forming a tubular surface of which the lines may be called the
elements. Then, in the first case, points moving on the tubular surface and cutting
the elements orthogonally will describe closed curves ; but, in the second case, they
will describe spirals.
The forces of a conservative system afford an example of the first or integrable
case. For, if X, Y and Z are the components, in the directions of the axes, of a
force whose direction and magnitude are functions of x, y and z, the lines of force
are those whose differential equations are
dx _ dy _ (h_
'X~ Y~ Z'
The equation
Xdx + Ydy + Zdz = o
will be satisfied by a particle moving perpendicularly to the lines of force, so that no
work is done upon it by the force ; and this equation is integrable, the integral
F= C being the equation of a system of kvel surfaces to which the lines of force are
everywhere normal.
284 EQUATIONS INVOLVING THREE VARIABLES. [Art. 261.
the point can move perpendicularly to the auxiliary lines. We
thus have a singly infinite system of lines on the given surface,
for the solution of the restricted problem.
262. In a general solution the assumed surface, as, for
example, the cylindrical surface represented by equation (i),
Art. 257, must be capable of passing through the line in space
representing any particular solution ; and, the surface being thus
properly determined, the line in question will be a member of
the singly infinite system determined upon the surface by the
additional integral equation found.
The peculiarity of the general solution of Art. 258 is that
the assumed surface V= <^(z) is made up of elements which are
themselves particular solutions of a certain class. We still have
a singly infinite system of particular solutions upon the assumed
surface, namely, the elements just mentioned. But upon each
surface there is in addition the unique solution determined by
equation (6). The points on the line thus determined are excep-
tions to the general rule, mentioned in the preceding article,
that at each point there is but one direction on the surface in
which a point can move perpendicularly to the auxiliary lines.
The line is, in fact, the locus of the points at which the auxiliary
lines pierce the surface orthogonally.
Examples XX.
Solve the following integrable equations : —
1. 2{y-{-z)dx +(x + sy + 2z)dy+ {x+y)dz = o,
{.x+yy{y+z) = c.
2. {y — z)dx + 2{x -\- 2,y — z)dy — 2{x + 2y)dz = o,
{x + 2y){y-zY = c.
3. {a — z) {^ydx -\- xdy) + xydz — o, xy=:c{z — a).
§XX.]
EXAMPLES.
285
4 {y-\-afdx + zdy—i^y-\-a)dz=o, z = {x ■\- c) {^y ■'f- a) .
5 . {ay — bz') dx+ {cz — ax) dy + {6x — cy)dz = o,
(ax — cz) = C{ay — bz) .
6. dx +dy + {x +y + z+ i)dz = o, {x+y + z)e^ = c.
7. {y'^ +yz)dx + {xz + z^)dy + {y^ — xy)dz = o,
y{x+z) = c(y + z).
8. {x' + z=) {xdx +ydy + zdz) + {x^ +y^ + s!')i{zdx — xdz) = o,
(;c=+_y= + z^)^+ tan-i- = <:.
z
9. 2(2)"^ +J'z — z')dx + x{4y + z)dy + x{y — 2z)dz = o,
a;2 (j[) -f- z) ( 2JC — z) = r.
10. (x^ji' —y3 —y'z)dx + (xy^ — a:3 — x'z)dy + (xv^ + a;=_)')^2 = o,
x+z , 7+g _ ,
11. {2X' + 2XV + 2X2^ + i)(& + dy + 2zdz = o,
^'^'(j' + z= + ^) + ^ = o.
12. (2;!!; -\- y + 2xt— z)dx + 2je)'(^ — ^Ci/z + x^dt = o,
x^ + xy* + x'^t — xz = c.
13. t{y + z)/& + /(jK + Z+ i)dy + /V/z — (j + z)dt = o,
(j/ + z)e^+y = ct.
14. z(j' + z)dx + z(« — x)dy ■\- y{x — «)«& +y{y + z)du = o,
(j + z){u + <:) + z{x — u) = o.
15. Find the equation which expresses the solution of
dz = aydx + bdy
when we assume y =f{x).
f{x)dx + 4/(a:) + C.
286 EQUATIONS INVOLVING THREE VARIABLES. [Art. 262.
16. Find the equation which determines upon the ellipsoid
h 4- H = I
a= b'^ c^
the lines which satisfy
xdx +ydy + cli -]dz = o.
x^ +^2 + z2= c.
1 7. Find the equations which determine upon the sphere
^2 .^y2 -I- Z2 = yj2
the lines which satisfy
\x{x — a) +y{y — b')\dz= {z ~c) {xdx +ydy).
z = C, and ax + dy + cz = k".
18. Show that, for the differential equation of Ex. 17, the auxiliary
system of lines consists of vertical circles, and verify geometrically the
results.
19. Give the general solution in Monge's form of the equation
zdx + xdy +ydz = o.
y + zlogx = 4>{z), x' (z) + y = X log X.
20. Find a general solution of
ydx = (x — z). (dy — dz) .
y-z={x), y = (x-z)4,'(x).
§ XXL] PARTIAL DIFFERENTIAL EQUATIONS. 287
CHAPTER XI.
PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER.
XXI.
Equations involving a Single Partial Derivative.
263. An equation of the form
Pdx -^ Qdy -\- Rdz = o (i)
which satisfies the condition of integrabihty is sometimes called
a total differential equation, because it gives the total differ-
ential of one of the variables regarded as a function of the
other two. Thus, if x and y be the independent variables, the
equation gives
dz = ---dx -%dy,
K K
or, in the notation of partial derivatives,
^=-^, (.)
dx R ^ '
and
%-%■■ «
that is to say, we have each of the partial derivatives of s given
in the form of a function of x, y and z.
288 PARTIAL DIFFERENTIAL EQUATIONS [Art. 263.
An equation of the form -(2) or (3), giving the value of a
single partial derivative, or more generally an equation giving
a relation between the several partial derivatives of a function
of two or more independent variables, is called a partial differ-
ential eqtcation.
264. To solve a partial differential equation of the simple
form (2), it is only necessary to treat it as an ordinary differential
equation between x and z, y being regarded as constant, and
an unknown function of y taking the place of the constant of
integration. The process is the same as that of solving the
total differential equation, see Art. 246, except that we have no
means of determining the function of y, which accordingly
remains arbitrary. Thus the general solution of the equation
contains an arbitrary function.
Equations of the First Order and Degree.
265. Denoting the partial derivatives ot zhy J> and g, thus
i» — — _ dz
dx dy
a partial differential equation of the first order, in which z is
the dependent and x and y the independent variables, is a rela-
tion between /, q, x, y and z. A relation between x, y and z is
a particular integral, when the values which it and its derived
equations determine for z, p and q in terms of x and y satisfy
the given equation identically. We shall find that, as in the
case of the simple class of equations considered in the preced-
ing article, the most general solution or general integral con-
tains an arbitrary function.
266. The equation of the first order and degree may be
written in the form
Pp + Qq = R, (i)
§ XXI. J OF THE FIRST ORDER AND DEGREE. 289
where P, Q and R are functions of x, y and z. This is some-
times called the linear equation, the term linear, in this case,
referring only to/ and q.
Let , .
u=^ a, (2)
in which u is a function of x, y and z, and a is a constant, be an
integral of equation (i). Taking derivatives with respect to
X and y, we have
du , du ^ J du , du
1 p = o, and 1 q — o\
dx dz dy dz
and substituting the values of p and q, hence derived in equa-
tion (i), we obtain
Tjdu . ^du , Tjdu . ,
"^^^^^^^^^^ (3)
Therefore, li it = a is an integral of equation (i), ?< is a function
satisfying equation (3),* and conversely.
But we have seen in Art. 231 that this equation is satisfied
by the function u when ti = a \a an integral of the system of
ordinary differential equations,
dx _ dy _ dz , ,
'P-Q-R ^4)
Hence every integral of the system (4) is also an integral of
equation (i).
Now, it was shown in Art. 232, that if
u — a and v = b
* It follows from the definition of an integral that this equation is either an
identity, or becomes such when z is eliminated from it by means of equation (2);
but, since it does not contain the constant a which occurs in equation (2), the former
alternative must be the correct one.
290 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 266.
are two independent integrals of the system (4), the equation
f{u, v) = C
includes all possible integrals of the system. Hence this equa-
tion, in which / is an arbitrary function, is the general integral
of equation (i). It is unnecessary to retain an arbitrary con-
stant since/ is arbitrary; in fact, solving for u, the equation may
be written in the form
u = {v),
which expresses the relation between x, y and z with equal
generality.
Thus, to solve the linear equation (i), we find two inde-
pendent integrals of the system (4) in the forms 11 = a, v = b,
and then put u = 4> {v), where <^ is an arbitrary function. This
is known as Lagrange s solution.
267. It is readily seen that we can derive in like manner
the general integral of the linear partial differential equation
containing more than two independent variables. Thus, the
equation being
=X, . . . . (l)
j^ p dz
dx„
the auxiliary system is
dx^ dx^
P. P. '"
dx„ dz
Pn R
(2)
and, if u-^ = c-,, u^ = c^, . . ., Un = c„ are independent integrals of
this system, the general integral of equation (i) may be written
/(«!, «2, ...«„) = o, (3)
where / is an arbitrary function. If an insufificient number of
integrals of the system (2) is known, any one of them, or an
equation involving an arbitrary function of two or more of the
quantities «i, 7/^, . . ., ti„ constitutes a particular integral of
equation (i).
§ XXL] THE LAGRANGEAN LINES. 29I
Geometrical Illustration of Lagrange'' s Solution.
268. The system of ordinary differential equations empxbyed
in Lagrange's process are sometimes called Lagrange s equations.
In the case of two independent variables they represent a doubly
infinite system of lines, which may be called the Lagrangean
lines. We have seen in Art. 235 that every integral of the
differential system represents a surface passing through lines
of the system, and not intersecting any of them. It follows,
therefore, that the partial differential equation
Pp->rQq=-R
is satisfied by the equation of every surface that passes through
lines of the system represented by Lagrange's equations
dx _ dy _ dz _
and the general integral is the general equation of the surfaces
passing through lines of the system.
Given, for example, the equation
{i7iz — ny)p -\- {nx — lz)q=- ly ^ mx, ... - (i)
for which Lagrange's equations are
dx dv dz / V
^ = := (2)
mz — ny nx — Iz ly — mx
The integrals of this system were found, in Art. 230, to be
Ix + my + nz = a,
and
^2 +^= + 2^ = 3;
and, as stated in Art. 236, the lines represented being circles
having a fixed line as axis, every integral of the system (2)
292 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 268.
represents a surface of revolution having the same line as axis.
Thus the general integral of equation (i), which is
Ix -'r my -\- nz =■ ^{x^ -\- yf -\- Z^') , (3)
represents all the surfaces of revolution of which the line
X _ y _z
I m n
is the axis.
269. It was shown in Art. 260 that, when
Pdx -\- Qdy -\- Rdz = o (i)
is the differential equation of a system of surfaces, the system
of lines represented by
dx dy dz , .
-P^Q=R (^>
cuts these surfaces orthogonally. It follows that the surfaces
represented by the general integral of
Pp+Qq = R,
which pass through the lines of the system (2), cut the surfaces
of the system (i) orthogonally. Hence, as first shown by
Lagrange,* if the equation of a system of surfaces containing
one parameter c be put in the form
V=c,
the surfaces which cut the system orthogonally are all included
in
* CEuvres de Lagrange, vol, iv. p. 628; vol. v. p. 560,
§ XXL] COMPLETE AND GENERAL PRIMITIVES. 293
where ?< = « and v = b are two independent integrals of
dx
dV
_ dy _
dV
dz
dV
dx
dy
dz
The Complete and General Primitives.
270. If, in an equation containing x, y and z, z be regarded
as a function of x and y, we may, by differentiation with respect
to X and J/, obtain equations involving/ and q respectively ; and
by the combination of the given and the two derived equations
we can derive a variety of partial differential equations satisfied
by the given equation. If the given equation contains two
arbitrary constants, their elimination leads to a definite differ-
ential equation of the first order independent of these constants,
and of this equation the given equation is called a complete
primitive.
Given, for example, the equation
s= a{x+y) +b (i)
By differentiation we have p = a, and q = a, hence
P = q (2)
is the only equation of the first order independent of a and b,
which can be derived from equation (i). Hence equation (i)
is a complete primitive of equation (2). We do not say the
complete primitive, because the general solution of / = ^ is
z=f{x+y), (3)
and therefore any equation of this form containing two arbitrary
constants is a complete primitive oi p = q. In fact, equation (3)
gives/ =f(x +y), q =f{x +y), whence p = q. The equation
294
PARTIAL DIFFERENTIAL EQUATIONS. [Art. 270.
from which a given partial differential equation can be obtained
by the elimination of an arbitrary function is called its general
primitive ; thus equation (3) is the general primitive oi p = q.
271. The most general equation between x, y and z, contain-
ing one arbitrary function, may be written in the form
/(«, ») = o, (i)
where u and v are given functions of x, y and z. Regarding z
as a function of x and y, the derived equations are
and
du
d£
du
du du '
dx dz
du du
dy dz
dv
dv
'dv__^dv_^
dx dz
]-"
dv .dv
dy dz
= o.
The result of eliminating the ratio -^ : -L may be written i
du dv
the form
in
du
dx
dv
■P
du
~dz
dv
du . du
dy dz
+ P
dx dz
dv
dy
dv
dz
Of the four determinants formed by the partial columns, that
containing /^ as a factor vanishes, and we have
du
du
du
du
du
du
dx
dv
dy
dv
+ P
dz
dv
dy
dv
+ ?
dx
dv
dz
dv
dx
dy
dz
dy
dx
dz
an equation of the form
Pp+Qg = R,
§ XXL]
THE GENERAL PRIMITIVE.
295
in which
P =
du
du
du
du
du
du
dv
dz
dz
dx
dx
dv
, e =
, R =
dv
dv
dv
dv
dv
dv
dy
dz
dz
dx
dx
dy
It thus appears that the equation of which the general primitive
contains a single arbitrary function is linear with respect to
p and q.
272. The values of P, Q and R above are called the Jacob-
ians of u and v with respect to y and z, s and x, x and y
respectively, and are denoted thus,
p ^ d{u, v) ^ ^ d{u, v) ^ ^ d(u, v) _
d{y, z) ' d{z, x) ' d{x, y) '
The Jacobian vanishes when u and v are not independent func-
tions of the variables expressed in the denominator, thus R
vanishes if either u or v \s a. function of z only. Again, P, Q
and R all vanish if u is expressible as a function of v. In this
last case equation (i) is, in fact, reducible to v = c, which con-
tains no arbitrary function.
When P, Q and R are given, the functions u and v must be
such that their Jacobians are proportional to P, Q and R.
Now, if we put
u ■= a
and
v = b,
we shall have
du J , du , , du ,
— dx + —dy + — dz = o,
dx dy dz
dv J , dv , , dv ,
— dx + — dy + --dz = o;
dx dy dz
296 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 2J 2.
whence, solving for the ratios dx : dy : dz, vf& have
dx _ dy _ dz
d{u, v) d{u, v) d{u, v)
d{y, z) d(z, x) d{x, y)
Hence we shall have found proper values of u and z' if u = a
and V = d are integrals of
dx _ dy _ dz_
¥~Q~ J? '
We have thus another proof of Lagrange's solution of the linear
equation.
273. In like manner, if there be n independent variables
Xj, x^, . . ., x„, and one dependent variable 0, we can eliminate
the arbitrary function/ from the equation
/{Ui, u^ . . . u„) = o,
in which «/„ u^, . . ., ii„ are n independent given functions of the
variables. In the result of elimination the coefficient of the
products of any two or more of the partial derivatives will
vanish, and we shall have an equation linear in these deriva-
tives, that is an equation of the form
/'i/i + P.A + . . . + Pnp^ = R.
Moreover, each of the coefficients P^, P^, . . ., P„ and R will
be the Jacobians of «i, ti^, . . ., tt„ with respect to n of the
variables, and the simultaneous ordinary equations derived from
li^ = Ci, U2 = C2, . . ., tin = c„ will be
dx-^ dx^ dx„ dz
P% -t 2 Pn P
where P^, P^, . . ., P„ and R are the same Jacobians.
;§ XXI.] EXAMPLES. 2g7
Examples XXI.
Solve the following partial differential equations : —
1. y 2X — 2Z — y =: o, X + y + z = y^4>(x)
dy
X
2. psiiy' -x^)=y, z=ywa.-^- + {y)
3. Ip + mq=i, z=j + 4>{ly—mx)
4. f + g = nz, z = e"y 4>{.x — y)
5. xp + yq = nz, z = x"(-
6. y/> + xq = z, z = (x +y){x' —y)
7. (y^x — 2x'>)p + (2_y+ — x^y) q = <)z{x? — J^),
8. xzp + yzq = xy, z^ = xy + 4>i ^
y"
9. x'^p - xyq + y = o, z:=—+{xy)
10. zp + yq = X, X + z= y4>{x^ — z^)
11. xp + zq + y = o,
12. {y + z)p+{z + x)q = x+y,
y
II. xp -\- zq -{■ y = o, tan-'- = loga: + <^(_>'^ +z=)
(z —y) sj{x +y + z) = ^
^A.^-y
X — z
nxy , V , , fy — X
13. x^p+rq^nxy, , = __log- + ix + y + z).
iS-/-?=-^^' (x+y)\ogz = x + ^ is a surface of revolution,
and find its axis.
24. If z^ = o and v = o are particular integrals of a linear partial
differential equation, show that every other integral <^ = o satisfies the
equation
'^('j>, «, v) _ Q
fl'(x, J, z)
25. Determine the surfaces which cut orthogonally the system of
similar ellipsoids „ , • 2\
=L+L + Z' = C\ <^(^, I-\^
m^ Tf \ z z
26. Determine the surfaces of the second order which cut orthogo-
nally the spheres , , , , ,
•' ^ x^ +y' -{- z' = 2ax.
x^ + y"" + !^ = 2by -\- 2CZ,
§ XXII.] EQUATIONS NOT OF THE FIRST DEGREE. 299
XXII.
The Non-Linear Equation of the First Order.
274. We have seen in Art. 270 that a partial differential
equation of the first order may be derived from a given primi-
tive by the elimination of two arbitrary constants. Such a
■ primitive constitutes a complete integral of the differential
equation ; but, when the resulting equation is linear, the general
solution contains an arbitrary function which imparts a gen-
erality infinitely transcending that produced by the presence of
arbitrary constants or parameters. The surfaces represented
by a complete integral constitute a doubly infinite system of
surfaces of the same kind, while the more general class of sur-
faces represented by the general integral is said to form a
family of surfaces. Thus, in the example given in Art. 270,
the complete integral (i) represents the doubly infinite system
of planes parallel to a fixed line ; and the general integral (3)
represents the family of cylindrical surfaces whose elements are
parallel to the same fixed line.
275. The differential equation derived from a complete prim-
itive may be non-linear. For example, if, in the primitive,
{x-hy ■\-(yy-kY + ^ = (^, (i)
h and k are regarded as arbitrary parameters, the resulting
differential equation is
z^C/^' + r + 1) =<^, (2)
which is not linear with respect to / and q. Equation (i) is
therefore a complete integral of equation (2). Geometrically it
represents a doubly infinite system of equal spheres having
their centres in the plane of xy. It will be shown, however, in
300 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 275.
the following articles, that the geometrical representation of the
general integral of a non-linear equation is a family of surfaces
equally general with that representing the general integral of a
linear equation. But, since it has been shown in Art. 271 that a
primitive containing an arbitrary function gives rise in all cases
to a linear equation, it is obvious that the general integral of a
non-linear differential equation cannot be expressed by a single
equation.*
The System of Characteristics.
276. A partial differential equation of the first order, con-
taining two independent variables, is of the form
F{x,y,z,p, q) = o (i)
Let
z= 4>{x,y), (2)
whence
/ = #. ^=#, (3)
dx dy
be All integral ; then these values of z, p and q satisfy equation
(i) identically. If x, y and s be regarded as the coordinates of
a point, equation (2) represents a surface. A set of correspond-
ing values of X, y, z, p and q determine not only a point upon
the surface, but the direction of the tangent plane at that point,
and are said to determine an element of the surface. If we per-
mit ;tr-and y to vary simultaneously in any manner, the corre-
sponding consecutive elements of surface determine a linear
* The surfaces of the same family are generated by the motion of a curve in
space, when arbitrary relations exist between its parameters. The simplest case is
that ill which there are but two parameters ; the two equations of the curve can then
be put in the form u = cj, "' — iTj ; and, if /(iTi, cj) = o is the relation between the
parameters, /(ii, v) = o is the general equation of the family. This case, therefore,
corresponds to the linear differential equation. See Salmon's " Geometry of Three
Dimensions," Dublin, 1874, pp 372 et seq.
§ XXII.] THE SYSTEM OF CHARACTERISTICS. ■ 30I
element of surface ; that is, a line upon the surface together
with the direction of the tangent plane at each point of the
line.
The linear element thus determined upon the surface (2) will
in general depend upon the form of the function i> ; but it will
now be shown that, starting from any initial point upon the
surface, there exists one linear element which is independent of
the form of <^, provided only that equation (i) is satisfied, so
that every integral surface which passes through the initial ele-
ment must contain the entire linear element.
277. Let the partial derivatives of i^be denoted as follows :
dF ^ dF „ dF _ .y dF _ r, dF _ ^
dx dy dz dp dq
Since z, p and q are functions of x and y, the derivatives of
equation (i) with respect to x and 7 give
x+z/ + /'^ + e^ = o, (4)
ax tijc
y + ^ + pf + <2j=o (5)
dy dy
Now let X and y vary simultaneously in such a way that
'^ = F, %=Q; (6)
dt dt
then, because for every point moving in the surface
dz = pdx + qdy,
we have also
dz
^=pP+qQ (7)
at
302 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 277.
Equations (6) and (7) give
dx _dy _ dz
'F~Q~pF+gQ
The values of p and q in these equations being given in terms
of X and J)/, by equations (3), they form a differential system for
the variables x, y and z. Starting from any initial point (xo,yo, ^o),
this system determines a line in space ; and, supposing the
initial point to be taken on the surface (2), this line lies upon
that surface.
Now, substituting from equation (6), and remembering that
dq _ d'^z _ dp
dx dxdy dy
equation (4) becomes
dx dt dy dt
whence
1=-^-^^ W
In like manner, equation (5) gives
f = -^-^^ (9)
Equations (6), (7), (8) and (9) now give
dx_ _dy _ dz _ '^_ Zoipo, q^, this system determines a linear element of
§ XXII. ] THE SYSTEM OF CHARACTERISTICS. ^O},
surface, and supposing the initial element to be taken on the
surface (2), the entire linear element lies upon that surface.
Now the system (10) is independent of the form of the func-
tion (^, and the only restriction upon the initial element is that
it must satisfy equation (i) ; it follows that every integral sur-
face which contains the initial element contains the entire linear
element. This linear element, depending only upon the form of
equation (i), is called a characteristic of the partial differential
equation. Through every element which satisfies equation (i)
there passes a characteristic*
278. A complete solution of the system (10) consists of four
integrals in the form of relations between x, y, z, p and q. Mul-
tiplying the terms of the several fractions by X, Y, Z, —P and
— Q, respectively, we obtain the exact equation dF=o, of which
F= C is the integral. But it is obvious that, in order to confine
our attention to the characteristics of the given equation, we
must take C=o. Thus the original equation is to be taken as
one of the integrals of the characteristic system. The other
three integrals introduce three arbitrary constants. Hence the
characteristics form a triply infinite system.
For example, in the case of the equation given in Art. 275,
which may be written
F = p' + q--'-+i==o, (i)
z=
X = o, F=o, Z=— , P=2p, Q = 2q, and the equations of
the characteristic are
* In like manner, when there are « independent variables, a set of values of
xj, Xi, . . ., x„, 2, /i, p2, . ■ ; pn, which satisfies the differential equation, is called an
element of its integral, and the consecutive series of elements determined as above
are said to form a characteristic. See Jordan's "Cours d' Analyse," Paris, 1887, vol
iii., pp. 318 et seq.
304 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 278.
dx _ dy _ dz _ _ z^dp _ _ z^dq , ,
P ^ p" + S' <^P c^q ' ' ' '
Of this system, equation (i) is an integral ; the relation be-
tween dp and dq gives a second integral which may be written
in the form
^=/tana _ . . . (3)
The values of/ and q derived from equations (i) and (3) are
\l(c^ — z^) , ,
p = cos a 1-^ L, (4)
z
q=svn.ay-S L, (5)
and these equations may be taken as two of the integrals, in
place of equations (i) and (3). Substituting these values in the
relations between dx and dy, dx and dz respectively, we obtain,,
for the other two integrals,
y = X tan o. + a, (6)'
and
{x %tca-\- bf = c^ — ^ (7).
These last equations determine, for given values of a, a and
b, the characteristic considered merely as a line, and then equa-
tions (4) and (5) determine at each point the direction of the
element, that is to say, the direction of a plane tangent to every-
integral surface which passes through the characteristic.
The General Integral.
279. It follows from Art. 277 that every integral surface
contains a singly infinite system of characteristics, so that if
we make the initial element of a characteristic describe an
§ XXII.] THE GENERAL INTEGRAL. 305
arbitrary line upon the surface (the linear elerhent of surface
along the line determining at each point the values of /o and q^,
the locus of the variable characteristic will be the integral sur-
face. Moreover, if we take an arbitrary line in ^space for the
path of the initial point, it is possible so to determine p„ and q^
at each point that the characteristic shall generate an integral
surface. For this purpose, we must have in the first place,
^(•«o, Jo, 2o, A, ?o) = o (i)
Again, since the path of the initial point is to lie in the surface,
so that
taking the differential equations of the arbitrary curve to be
dxa _dyo _ dza , ^
~L~M-N' ('^
we must have
■ N = PoL+qJi, (3)
where L, M and N are functions of x^, jTo and z^. Geometrically,
this last equation expresses the condition that the initial ele-
ment must be so taken that the plane tangent to the surface
shall contain the line tangent to the arbitrary curve.
The general integral may now be defined as representing
the family of surfaces generated by a variable characteristic
having its motion thus directed by an arbitrary curve.*
* That the surface thus generated is necessarily an integral will be seen in the
following articles to result from the existence of a complete integral. The analytical
proof requires that it be shown that, for a point moving in the surface, we have always
dz=pdx -f- qdy,
where / and q are given by the equations of the characteristic. If the common
value of each member of the equations (z) be denoted by dr, the variation of t moves
306 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 279.
In the case of the linear equation, when the characteristics
become the Lagrangean lines, the values of p^ and q^ are still
those which satisfy equations (i) and (3) ; but they need not be
considered, because there is but one Lagrangean line through
each point.
Derivation of a Complete Integral from the Equations of the
Characteristic.
280. The four integrals of the characteristic system contain
X, y, z, p, q, and three constants. We may therefore obtain, by
elimination if necessary, a relation between x,y, s and two of the
constants. Every such equation represents, for any fixed values
of the constants, a surface passing through a singly infinite sys-
tem of characteristics, but not in general a system of the kind
considered in Art. 279, so that the equation is not in general an
integral of the partial differential equation. It will now be
the characteristic, and that of t \dt being, as in Art. 277, the common value of each
member of equations (10)] moves a point along the characteristic. The motion of a
point along the surface then depends upon the two independent variables t and t.
Then, since
dz=^dt^-^dr, dx='^-^dtJr^dr,
dt dr dt dr
dy =
-.±dt+^dr.
dt dr '
and the equations of the characteristic give
dt ^ dt ^ dt
it remains only to prove that
dz ^dx , dy
— = / f- ? ^
dr dr dr
or that
dr dr ^ dr
Letting t=o correspond to the initial point, the condition dzo — fiadxc,.^ qadyo shovfS
that the corresponding value of U is zero, that is U^ = o. Consider now the value
dp dx _ d^x _d£dy_ d'^y
'dt dr dtdr dt dr dtdr'
dt
This is
dU dH
dt dtdr
§ XXII.] DETERMINATION OF A COMPLETE INTEGRAL. 307
shown how we may find such an integral, that is to say, since
two arbitrary constants occur, a complete integral of the given
equation.
Suppose one integral of the characteristic system, in addition
to the original equation /^= o, to have been found. Let a denote
the constant of integration introduced, and consider the values
of p and q in terms of x, y, z and a determined by these equa-
tions. Now, in a complete solution of the characteristic system,
each characteristic is particularized by a special value for each
of the three constants of integration. We may distinguish those
in which a. has the special value a^, as the aj-characteristics ;
these constitute a doubly iniinite system of linear elements of
surface, which together include all the point elements deter-
mined by the above-mentioned values of p and q, when the par-
ticular value ttj is assigned to a.
Now these ai-characteristics lie upon a system of integral
surfaces. To show this, consider a transverse plane of refer-
But
dH__d_ dz_dp dx (i'x dq dy d'^y _
dtdr ~ dT dt~ dr dt drdt dr dt drdt '
hence
dU _d^ dx d^ ^ _dp dx _d£_ ^
dt ~ dT dt dr dt dt dr dt dr
Substituting from the equations of the characteristic, this becomes
dt dr dr dr dr dr dr
or, since Zdz + Xdx -f- Ydy + Pdp J^ Qdq = o,
dU^_
dt
The integration of this gives
dU 7<& , ^^dx , „Tdy y,,
--=-Z—-|-/Z — + qZ-f = -ZU.
dt dr dr ar
and, putting /= o, we have C= ^/^ = o ; hence, so long as the exponential remd -s
finite i/= o, virhich was to be proved. See Jordan's " Course d'Analyse," vol. iii.,
p. 323-
308 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 280.
ence. This is pierced at each point by one of the a,-characteris-
tics, and at the point the element, which we may take as the
initial element of the characteristic, determines in the plane of
reference a direction. If, starting from any position in the
plane of reference, the initial point moves in the direction thus
defined, it describes a determinate curve in that plane, and
the corresponding characteristic generates an integral surface.
Varying the initial position in the plane of reference, we have a
singly infinite system of curves in that plane, and a singly infi-
nite system of integral surfaces.
We have thus a system of surfaces at every point of which
the values of/ and q are the values above mentioned which
involve u-^. Hence, if these values be substituted in the equa-
tion
dz ■=pdx -|- qdy
(which, it will be noticed, is, by Art. 277, one of the differential
equations of the characteristic system), we shall have an equa-
tion true at every point of this system of surfaces ; in other
words, we shall have the differential equation of the system.*
The integral of this equation will contain a second constant
of integration's; when both constants are regarded as arbitrary,
it represents a doubly infinite system of surfaces containing the
entire system of characteristics, and is a complete integral.
281. As an illustration, let us resume the example of Art.
278. Substitution of the values of / and q, equations (4) and
(5), in dz ==pdx -f qdy, gives
zdz r , , .
= dx cos a + dy sin a.
y/(f2 _ 22)
* It follows that the equation thus found is always integrable. This would, of
course, not be generally true if the values of p and g simply satisfied the equation
F=o, The early researches in partial differential equations were directed to the
discovery of values of p and ? which satisfied F= o and at the same time rendered
dz = pdx + qdy integrable. See Art. 294.
§ XXII.] DETERMINATION OF A COMPLETE INTEGRAL. 309
whence, integrating, we have
z'^ + {x cos a + 7 sin a + Py- = c%
which is therefore a complete integral of the given equation
z=(/= + ?= + i) = c'^-
This complete integral represents a right circular cylinder of
radius c, having its axis in the plane of xy ; and since equation
(6), Art. 278, represents a plane perpendicular to the axis, we
see that the characteristics in this example are equal vertical
circles, with their centres in the plane of xy, regarded as elements
of right cylinders.
It follows that the general integral represents the family of
surfaces generated by a cijcle of radius c, moving with its centre
in, and its plane normal to, an arbitrary curve in the plane of
xy. The surfaces included in the complete integral just found are
those described when the arbitrary path of the centre is taken
^ a straight line.
Relation of the General to the Complete Integral.
282. Since all the integral surfaces which pass through a
given characteristic touch one another along the characteristic,
and the surfaces included in a complete integral contain all the
characteristics, it follows that every integral surface touches at
each of its points the surface corresponding to a particular pair
of values of a. and /? in the equation of the complete integral.
The series of surfaces which touch a given integral surface cor-
responds to a definite relation between /8 and a, say /? = <^ (a) ;
thus the given integral is the envelope of the system of surfaces
selected from the complete integral by putting ^=<^{a) and so
obtaining an equation containing a single arbitrary parameter.
3IO EQUATIONS NOT OF THE FIRST DEGREE. [Art. 282
The equation of the envelope of a system of surfaces repre-
sented by such an equation is found in the same manner as that
of a system of curves. See Diff. Calc, Art. 365. That is to
say, we ehminate tlie arbitrary parameter from the given equa-
tion by means of its derivative with respect to this parameter.
283. For example, in the, complete integral found in Art
281, if a and /8 are connected by the relation
/? cosa + ^ sina + /8 = o, (i'
the equation becomes
^ -\-\{x — K) co%a.-\-{^y — k')s\a.a\'' = C". . . . (2'
Taking the derivative with respect to u,, we obtain
[(x — k) cos a + (jv — ^) sin a] [(ji/ — kj cos a — {x — A) sin a] = o,
whence we must have either
(jc — A) cos a + (jc — ^) sin a = o, . • . . (3)
or else
(jc — ,^) cos a — (.« — ^) sin a = o (4)
The elimination of a from equation (2) by means of equation (3)
■^ives
2" = ^, (5)
and, in like manner, from equations (2) and (4) we obtain
z''+ (x — hy -\- (j — Kf^c^ (6)
Equation (i) expresses the condition that the axis of the cylin-
der represented by the complete integral shall pass through the
fixed point {h, k, o) ; accordingly the envelope of the system (2)
consists of the planes z=±c, and the sphere (6) whose centre is
§ XXII.] EXPRESSION OF THE GENERAL INTEGRAL. 31 1
{h, k, o). Regarding h and k as arbitrary, equation (6) is tlie
complete integral from which as a primitive the differential
equation was derived in Art. 275.
284. To express the general integral, the relation between
the constants in the complete integral must be arbitrary. Thus,
the complete integral being in the form
f{x,y,z, a,b) = o, (i)
we may put b = ^ (a), where denotes an arbitrary function, and
then the general integral is the result of eliminating a between
the equations,
/[x,y,z,a,^{a)^ = o, (2)
and
^ /[x, y, z, a, ct^{a)^ = o (3)
da
The elimination cannot be performed until the form of ^ is spe-
cified ; for, as remarked in Art. 275, the general integral cannot
be expressed by a single equation unless the given partial differ-
ential equation is linear.
Since the general integral can thus be expressed by the aid
of any complete integral, we shall hereafter regard a non-linear
partial differential equation as solved when a complete integral
is found.
Singular Solutions.
285. There may exist a surface which at each of its points
touches one of the surfaces included in the complete inte-
gral without passing through the corresponding characteristic.
Every element of such a surface obviously satisfies the differen-
tial equation, and its equation, not being included in the general
integral, is a singular solution analogous to those which occur
in the case of ordinary differential equations.
312 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 285.
An integral surface generated, as in Art. 279, by a moving
characteristic will in general touch the surface representing the
singular solution along a line. If the surfaces of the complete
integral have this character, the singular solution will be a part
of the envelope found by the process given in the preceding-
article, no matter what the form of <^ may be. In this case, equa-
tions (2) and (3), Art. 284, which together determine the ultimate
intersection of consecutive surfaces of the system (2), represent
a characteristic and also the line of tangency with the singular
solution. The former, as a varies, generates a surface belong-
ing to the general integral, and the latter generates the singular
solution. Thus, in the example of Art. 283, equation (3) deter-
mines upon the cylinder (2) its lines of contact with the planes
^r = ± c, and equation (4) determines a characteristic.
286. There is, however, when a singular solution exists, a
special class of integrals which touch the singular solution in
single points, each of these being in fact the envelope of those
members of the complete integral which pass through a given
point on the singular solution. This class of integrals obviously
constitutes a doubly infinite system, and thus forms a complete
integral of a special kind. The complete integral (6), Art. 283,
is an example.
When f(^x, y, z, a,b)=o
is the complete integral of this special kind, the characteristics
represented by equations (2) and (3), Art. 284, will, for given
values of a and b, all pass through a common point, indepen-
dently of the form of <^, and this point will be upon the singu-
lar solution. In particular, the characteristic defined by _/= o
and ^ = will intersect that defined by y"=o and J- = o,
da db
in a point on the singular solution. Hence, in this case, the
singular solution will be the result of eliminating a and b from
t;he three equations.
§ XXII.] SINGULAR SOLUTIONS. ' 313
/=o, -^ = O, -J-=o.
•'da db
It is to be noticed, however, that the eliminant of these
equations may, as in the case of ordinary differential equations,
include certain loci which are not solutions of the differential
equation.
287. Since the characteristics which lie upon a surface of the
kind considered above, all pass through the point of contact with
the singular solution, it follows that the singular solution is the
locus of a point such that all the characteristics which pass
through it have a common element. At such a point, therefore,
the initial element fails to determine the direction of the charac-
teristic. Now, in the equations (10), Art. 277, the ratio dx:dy
is indeterminate only when P = and Q = o, or when P = 00
and S = °° ; hence one of these conditions must hold at every
point of a singular solution. The former is the more usual case,
so that a singular solution generally results from the elimination
of/ and q from
F{x,y, 2, A q) =0
by means of the equations
dF , dF
= o and = o.
dp dq
It is necessary, however, to ascertain whether the locus thus
found is a solution of the differential equation, for the conditions
p _ o, 2 = 0, and P =00, 2 = 00 are satisfied at certain other
points besides those situated upon a singular solution ; for ex-
ample, those at which all the characteristics which pass through
them touch one another. In the example of Art. 278, P = o,
(2 = o gives the singular solution c = ± c, and P = 00, Q — co
gives z' = o, which is the locus of the last-mentioned points,
and not a solution.
314 EQUATIONS NOT OF THE FIRST BEGREE. [Art. 28?.
Equations Involving p and q only.
288. We proceed to consider certain cases in which a com-
plete integral is readily obtained. In the first place, let the
equation be of the form
F{p, ?) = o (i>
In this case, since X=o, F=o, Z=o, two of the equations
[(10), Art. 277] of the characteristic become dp = o and dq = o\
whence
p =^ a and q ■= b (2)
The constants a and b are not independent, for, substituting in
equation (i), we have
F{a, ^) = o (3)
Substituting in dz =pdx + qdy, we obtain
dz = adx + bdy ;
whence, integrating, we have the complete integral
z = ax -\- by -\- c, (4)
where a and b are connected by equation (3), and c is a second
arbitrary constant.
289. The characteristics in this case are straight lines, and
the complete integral (4) represents a system of planes. The
general integral is a developable Surface. There is no singular
solution.
A special class of integrals which may be noticed are the
envelopes of those planes belonging to the system (4) which
pass through a fixed point.* These are obviously cones, whose
* The characteristics which pass through a common point in all cases determine
an integral surface. The integrals of this special kind constitute a triply infinite
system ; we may limit the common point or vertex to a fixed surface (as, for example,
in Art. 286, to the singular solution), and still have a complete integral.
§ XXII. J EQUATIONS ANALOGOUS TO CLAIRAUT'S. 315
elements are the characteristics which pass through the fixed
point. For example, if the equation is
p^ + g^ = m",
these cones are right circular cones with vertical axes, and their
equations are
{z — y)^ = nf^(x — a)' + w(y — ^Y-
Equation Analogous to Clairaut's.
290. There is another case in which the characteristics are
straight lines ; namely, when the equation is of the form
z
= px + qy+f{p,q) (i)
In this case, X =p, F= q, Z = — i, and we have again, for two
of the equations of the characteristic, dp = and dq = o\ whence
p= a, q = b (2)
Substituting in dz =pdx + qdy, and integrating, we have the
complete integral
z = ax + by + c, (3)
in which the constant c is not independent of a and b ; for, sub-
stituting the values of / and q, equation (i) becomes
z = ax-Vby -\-f{p->b), (4)
which, since it is also one of the integrals of the characteristic
system, must be identical with equation (3).
291. The complete integral in this case also represents a
system of planes, and the general integral is a developable sur-
face. A singular solution also exists.
3j6 equations not- of the first degree. [Art. 291.
For example, let the equation be
z = /^ + 2y + ,^V^(i +/=■+?='); (i)
the complete integral is
z= ax ->rby-\-k^{i -\- a'^ -^b'^) (2)
For the singular solution, taking the derivatives with respect to
a and b, we have
X H ; = O,
and
V -i = O.
^^ ^(^i + a^ + d^)
These equations give
a = — , b =
v/(^^ — x" —yy ~ \J{k' — x^ — /^)'
and, substituting in equation (2), we have
x" + f + z:^ = k' (3)
Thus the singular solution represents a sphere, the complete
integral (2) its tangent planes, and the general integral the
developable surface which touches the sphere along any arbi-
trary curve.
Equations not Containing x or y.
"2,^1. When- the independent variables do not explicitly occur,
the equation is of the form
F{z,p,g) = o (i)
§ XXII.] EQUATIONS OF SPECIAL FORMS. 317
Here X=o and Y=o, and the final equation of the character-
istic system reduces to
dp dq
whence
q = ap (2)
Substituting in equation (i), we have F{z,p,ap) =0, the solu-
tion of which gives for / a value of the form
Thus, ds =pdx + qdy becomes
dz = (jl)(z) {dx + ady) ;
whence, integrating, we have the complete integral,
x-it-ay = f— ^ +b (3)
The illustrative example of Arts. 278 and 281 is an instance
of this form. It will be noticed that the mode of solution leads
to a complete integral representing cylindrical surfaces whose
elements are parallel to the plane of xy. The equation
F{z, o, o) = o,
representing certain planes parallel to the plane of xy, will obvi-
ously be the singular solution.
Equations of the Form f-,{x,p) =fi{y, q).
293. When the equation does not explicitly contain z, it may
be possible to separate the variables x and p from y and q, thus
putting the equation in the form
Mx,p)=My,q) (i)
3l8 , EQUATIONS NOT OF THE FIRST DEGREE. [Art. 293.
In this case, we have Z=o, X= — , P= ~, and the equations
dx dp
of the characteristic give for the relation between dx and dp,
^ax + ^dp = o.
ax dp
Integrating, we have/i (jr,/) =«, and from equation (i),
A{x,p)=f^{y,q)=a (2)
Solving these equations for/ and g, we have values of the form
p = ^{x,d), i = 'i>^{y,a),
and dz =pdx + qdy becomes
dz = <})i(x, d)dx + ^2(j', ci)dy,
whence we derive the complete integral,
z= <^T,{x, a)dx -V- 4>2{y, a)dy + l>.
For example, let the given equation be
xp' + y^ = I •
Putting
xp^ = I — yq' = a,
we have
^ = $- ^-'-^'
and, integrating dz =pdx + gdy, we obtain the complete integral
z = 2\/a\/x + 2y/(i — a)\/Y + i.
§ XXII.] CHANGE IN THE CHARACTERISTIC EQUATIONS. 3 19
Change of Form in the Equations of the Characteristic.
294. If we make any algebraic change in the form of the
equation
F{x,y, z,p, q) = o,
the equations of the characteristic (10), Art. 277, will be altered.
The changes, however, will be merely such modifications as
might be produced by means of the equation F= o itself.* In
particular, the form assumed when the equation is first solved
for g may be noticed. Suppose the equation to be
g = (x,y,z,p), whence X=-^, V=-^, Z = -^,
ax dy dz
P = r, and Q=\. Putting q in the place of <^ in the partial
dp
derivatives, and omitting the member containing dq, the equa-
tions of the characteristic become
dx , dz dp , .
= dy = r=-, ^— T' • • . . (2)
dp ^ -^ dp dx ^ dz
a complete system for the four variables x, y, z and/, q being
the function of these variables, given by equation (i). These
equations may be deduced from the consideration that the val-
ues oi p and q derived from one of their integrals combined with
equation (i) should render dz=pdx +qdy integrable.f
* The complete solution of the characteristic system involving four arbitrary
constants (see Art. 278) would indeed be changed, but not the special solution in
which F= o is taken as one of the integrals.
t See Boole's "Differential Equations," London, 1865, p. 336.
320 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 295,
295. As an illustration, let us take the equation
z=pq, or <1=\ (^)
P
Equations (2) of the preceding article become
i- = dy = ^ — = dp (2)
z 22
Of these the most obvious integral is
P=y + a;
whence dz =pdx + qdy becomes
dz = ( V + a)dx -\ ^,
y + a
from which we derive the complete integral
z={y + a){x + b) (3>
The equations of the characteristic derived from the more
symmetrical form of the equation
F ^ pq ~ z ^= O
are
dx _dy _ dz _dp _dq , ^
q~ p~ 2pq~ p~ q'
which are readily seen to be equivalent to equations (2). If the
final equation of the system (4) be used, as in the process of
Art. 292, to determine/ and q, we shall have
/ = — , q = a\Jz,
giving
Az = (^ + ay + pJ, (S>
another complete integral of the equation z =pq.
§ XXII.] TRANSFORMATION OF THE VARIABLES. 32 1
Transformation of the Variables.
296. A partial differential equation may sometimes be re-
duced by transformation of the variables to one of the forms for
which complete integrals have been given in Arts. 288, 290, 292
and 293. The simplest transformation is that in which each
variable is replaced by an assumed function of itself. The
choice of the new variable will be suggested by the form of
the given equation.
Let
|=<^(x), V = "/'(>'). C=/(2),
then
dl =f{z)dz =f\z) f-'^^ + f dri\
'{x)dx ^^Kl,'{y)dy '■
Hence, denoting the partial derivatives of ^ with respect to
i and 17 by /' and q', their expressions in terms of x, y arid z are
the same as if they were ordinary derivatives.
For example, the equation
.ar^/^ + j'2^= = 2= (i)
may be written
fxdzV- fydz\^_
\zdxj \zdyj
Putting — = ^1, J^= d-q, — = di,, whence ^ = log;r, 17 = log 1/
X y z
and I = log z, the equation becomes
/'" + ?'"= I (2)
The complete integral of this equation is, by Art. 288,
^ = ai + 6r] + c,
322 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 296.
\
where a= + 3^ = i ; hence, putting a = cos a, b = sin a, the com-
plete integral of equation (i) is
log z = cos a log ;c + sin a logjv + c,
or
z = Cr'=°=''_y^'°''.
297. In the following example the new independent variables
are functions of both of the old ones. Given
Using the formulae connecting rectangular with polar coordi-
nates,
whence
we have
dz dz ■ n , dz cos 9
q = — = — smfc^H — - .
^ dy dr dO r
Substituting, equation (i) becomes
or, putting dp = — ,
Hence the integral is
= p cos a. + 6sva.a + p
y
= \ cos a log (x'' -\- y^^ + sin a tan-' — f- p.
The same complete integral may be found directly by the method
of characteristics (see Ex. 20).
X =■ r cos fi,
7 = r sin 9,
r^ = x= -\-y^.
e = tan-J
, _dz _ dz
^~~dx~~dr
cos
e
dz sin
dd r '
XXII.] EXAMPLES. 323
Examples XXII.
Find complete integrals for the following partial differential
y
z := ax -i \- }.
a
z = \{2X — a)^ + ay + i.
z = X sec a + y tan oi + b.
s =: ax -{- by + ab ;
singular solution, z = — xy,
z = axe^ + i a'e^y + b.
z = \ ax'^ -^ \ {a'' ~ i)^y + b.
z ^l{x + a)- ^ l{y - af + b.
z ■= ax ■\- a^y + b.
z-= ax -\- by — na^b" ;
I
singular solution, z = {2. — n) (x;')^^"-
z?^ (x Ydf-\- {y + a)^ + b.
yz = ax + 2 s/it^y) + b.
12. /= + ?^ — 2j>x — 2qy + 1 = o,
2Z = X' +y' + x\/{x^ + a) +y\/{y' — I — a)
[x + ^ix^ + a)T ^,_
13. Denoting x + ay by /, find a complete integral of Ex. 12 in
the form
14. (p + q)i^x + qy)= I, \/(i+a)z=2\/(x \-ay) +b.
equations
—
I.
/?
= I,
2.
\'P + \lq= 2x,
3-
P-.
-^ = 1,
i.
z =
px + gy+pq
5-
q =
-. xp +/%
6.
y\p-
— x'q^ = x^jc^,
7-
P'^-\-^ = X +y,
8.
i =
= 2yp^,
9-
z =
px + gy - npng",
10.
p--
X —y
-^= z '
II.
p =
■.{gy + zY,
324 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 29/.
OC^ I
16. x''y^z~^p'g = I, log— =
bz zwy^
17. p^ — y^q — y^ — x^,
s = — sm-i - -\ — i-!^ y + l>-
2 a 2 y
18. Find three complete integrals of
fig = px + qy.
1° 2Z = (^ + a^\p.
2° z = xy + y ^{x^ — a") + b.
3° z = xy + x\j{y'' + a'^) + b',
19. Show directly, by comparison of the values of z, / and q, that a
surface included i^ the integral 2" can be found touching at any givep
point a given surface included in the integral 1° ; and that the relation
K = 2O
will then exist between the constants. Hence derive one integral from
the other, as in Art. 283. Also show that the similar relations for the
other pairs of integrals are
P = 2b' + a'=a=, and b — b' = aa'.
20. Show that xq — yp= a is an integral of the characteristic system
for the equation
(*-+r) (/= + ?») = 1;
and thence derive the complete integral given in Art. 297.
21. Solve, by means of the transformations xy = ^, x +y = v- the
equation
(y - x) {qy -px) ={p- qy.
z = axy + ^{x -\-y) + b.
22. {x" — y)pq — xy(J>^ — q"") = 1.
z = ia\og{x^ +y) +lt3.n-^l+ b.
& X
§ XXII.] EXAMPLES. 325
23. Show that the equations of the characteristic passing through
(a, j8, y) in the case of the equation
> p^ -\- g^ = ni^,
Art. 289, are
x — a._y — j8 z — y
a b m'^
where a'^ -\- b^ = nt^ ; and thence derive the special integral given in
that article.
24. Deduce, in like manner, the integral formed by characteristics
passing through (h, k, /) for the equation
p^ -\- g^ = I.
{x — hy + {y — ky = l\/{£^ — /") —sj{c^ — z^)]=-
25. Show that when the complete integral is of the form
au -\- bv -^ w ^= o, (i)
where u, v and w are rational functions of x, y and z, the elimination
can be performed, giving the general integral
<^f^, ^) = o, (2)
\w WJ
a homogeneous equation in u, v, w. Accordingly, show that the equa-
tion arising from equation (i) as a primitive is the linear equation
Pp -\- Qq = R, where
d{y, z) d{y, z) d{y, z)
U 7)
with similar expressions for Q and R, and that putting «i = — , ^i =— ,
these values of P, Q and R agree with those derived from the general
primitive in Art. 271. >■
326 EQUATIONS OF THE SECOND ORDER. [Art. 298.
CHAPTER XII.
PARTIAL DIFFERENTIAL EQUATIONS OF HIGHER ORDER.
xxni.
Equations of the Second Order.
298. We have seen that the general solution of a partial
differential equation of the first order, containing two independ-
ent variables, involves an arbitrary function, although it is not
possible to express the solution by a single equation except
when the differential equation is linear with respect to p and q.
We might thus be led to expect that the general solution of an
equation of the second order could be made to depend upon two
arbitrary functions. But this is not .generally the case. No
complete theory of the nature of a solution has yet been devel-
oped, although in certain cases the general solution is expressi-
ble by an equation containing two arbitrary functions. We shall
consider these cases in the present section, and in the next, the
important class of linear equations with constant coefficients,
for which in some cases a solution of the equation of the nth.
order containing n arbitrary functions can be obtained.
TTie Primitive containing Two Arbitrary Functions.
H
299. If we consider on the other hand the question of the
differential equation arising from a given primitive by the elimi-
nation of two arbitrary functions, we shall find that it is only in
§ XXIII.] TWO ARBITRARY FUNCTIONS. 327
certain cases that the elimination can be performed without
introducing derivatives of an order higher than the second.
The general equation containing two arbitrary functions may
be written in the form
f\_x,y,z, {u), i/r(w)] = o,
in which u and v are* given functions of x, y and z. The two
derived equations
df ^ df
ax ay
will contain ^'(«) and i/''(z'), two new unknown quantities to be
eliminated. There will be three derived equations of the second
order
d^f d^f d^f
dx' dxdy dy^
containing two new unknown quantities, ^"(?/) and ^'\v). We
have thus in all six equations containing six unknown quantities.
The elimination, therefore, cannot in general be effected.*
300. Suppose, however, that the original equation can be put
in the form
w=<^{u) +«/'(»); (i)
then the two derived equations of the first order,
dw . dw , ,\f„\(du du \ . ,,,\(dv , dv \ , .
dw , dw •u.,\(du . du \ . ,,, .(dv , dv \ , .
_+_,= <^(.)(^-+-,j + ^'(.)(^_ + _,j,. . (3)
are independent of <^ and i/f. These, with the three derived
* If we proceed to the third derivatives, we shall have ten equations and eight
quantities to be eliminated, so that two equations of the third order could be found
which would be satisfied by the given primitive.
328 EQUATIONS OF THE SECOND ORDER. [Art. 30O.
equations of the second order, will constitute five equations
containing the four quantities <^', l/{u), (i)
in which we have two arbitrary functions of the same given
function of x, j/ and s. In this case the derived equations take
the form
'^)-«"(f)+'*'«(l)+*<"(l)" ■«
in which (-5-), etc., are written in place of — + —j>, etc.
\axj dx dz
Multiplying equations (2) and (3) by ( — j and (— j respectively,
and subtracting the results, ^\u) and \\i\u) are eliminated to-
gether, and we have again an intermediate equation of the first
order containing one arbitrary function.*
* The cases considered in this and the preceding article are not the only ones
in which an intermediate equation of the first order can arise. See, for instance, the
example given in Art. 311.
§ XXIIL] THE INTERMEDIATE EQUATION. 329
The Intermediate Equation of the First Order.
302. The preceding articles indicate two cases in which an
intermediate equation of the first order may arise from a primi-
tive. We have now to consider, on the other hand, the form of
the differential equations arising from an intermediate equation
of the form
« = (»). (i)
where ti and v now denote given functions of x, y, s, p and q,*
and <^ is an arbitrary function. Denoting the second derivatives
of £• by r, J and t, thus
d-'z d^'z
dx^ dxdy
d^z
~~ dy
the two derived equations are
du , du . , du , du ,,, ^ fdv , dv . , dv .dv
<^'(.) f^
-J
dx dz^ dp dq ^^'\dx dz^ dp ^ dq
du , du , du , du . ,, ., s fdv ,(y) +i'{y),
or, putting {y) in place of the function i<^(jj')— i^,
z = ^yx" \ogx + x^cl>{y) + ,j/(y).
305. Again, an equation which does not contain t may be
exact * with reference to x, y being regarded as constant. Given,
for example, the equation
/ + r + s = I J
integrating, we have
z+p + g = x + {y).
* The equation might also be such as to become exact with respect to the four
variables /, q, z and x, by means of a factor. For this purpose three conditions of
integrability would have to be satisfied; see Art. 252. This is the number of con-
ditions we should expect, since by Art. 303 two must be fulfilled to render an inter-
mediate integral possible, and one more is necessary to express that in that integral
v=^ y.
332 EQUATIONS OF THE SECOND ORDER. [Art. 305.
For this linear equation of the first order, Lagrange's equations
are
dx ■= dy ^ ,
x-z-\-^{y)
of which the first gives
X — y = a,
and this converts the second into
dz
J +z = a+y + ^{y),
of which the integral is
gyz = ae^ +
\_y + Hy)Y'^y + i>-
Hence, making b = ^{a), we have for the final integral
eyz = eyx — eyy + [[.y + (y)]eydy + ^{x —y),
or, with a change in the meaning of <^,
z = X + {y) + e~y^{x —y).
Mongers Method.
306. The general method of deriving an intermediate equa-
tion where one exists is based upon a mode of reasoning similar
to the following method for Lagrange's solution of equations of
the fifst order, which is that by which it was originally estab-
lished.
Given the equation
Pp + Qq = R, (i)
and the differential relation
dz = pdx + gdy, (2)
§ XXIII.] MONGE'S METHOD. 333
which must exist when ^^ is a function of x and y. Let one of
the variables/ and q be eliminated, thus
dy
or
p{Pdy - Qdx) + Qdz - Rdy = o (3)
Hence, the relation between x, y and z which satisfies equation
(i) must be such that, when one of the two differential expres-
sions occurring in equation (3) vanishes, the other will in general
also vanish. Let us now write the equations
Pdy-Q.dx=:^o\
Qdz — Rdy = o J '
and suppose u = a, v — b,\.o \>& two integrals of these simulta-
neous equations. Then du = o and dv =0 constitute an equiva-
lent differential system, and the relation between x, y and z is
such that, if du = o, then dv = o; that is, if ti is constant, v is also
constant. This condition is satisfied by putting
u = {v),
which is therefore the solution of equation (i).
Geometrically the reasoning may be stated thus : If upon a
surface satisfying equation (i) a point moves in such'a way that
Pdy — Qdx = o, then also will Qdz — Rdy = o ; that is, the point
will move in one of the lines determined by equations (4). No
restriction is imposed upon the surface, except that it shall pass
through these lines, namely, Lagrange's lines defined \)y u = a,
V = b. The general equation of the surface so restricted is
u = {y).
307. Monge applied the same reasoning to the equation
Rr + Ss + Tt=V, (i)
334 EQUATIONS OF THE SECOND ORDER. [Art. 307.
where R, S, T and Fare functions of x, y, z, p and q, in connec-
tion with which we have, for the total differentials of/ and q,
dp — rdx + sdy, (2)
dq = sdx + idy (3)
Eliminating two of the three variables r, s, t, we have
^ dp — sdy ^ ^j ^ ^ dq — sdx ^ y
dx dy
or
Rdpdy + Tdqdx—Vdxdy=s{Rdy^ — Sdxdy+TdX'). . .(4)
If, then, we can find a relation between x, y, z, p and q, such
that, when one of the two differential expressions contained in
equation (4) vanishes, the other will vanish also, this relation
will satisfy equation (i).
Let us now write the equations
Rdy^ — Sdydx + Tdx'' = q\
\ (s)
Rdpdy + Tdqdx = Vdxdy j
li u^^a and v = b are two integrals of this system, so that du = o,
and dv = o form an equivalent differential system, the required
relation will be such that if du — o, then dv = 0; that is, if u is
constant, v is also constant. As in the preceding article this
condition is fulfilled by
u = ^(»),
which is now a differential equation of the first order. The
integral of this equation is therefore a solution of equation (i).*
* The same method applies to the more general form (3), Art. 302, when an
intermediate integral exists, but the auxiliary equations are more complex. See
Forsyth's Differential Equations, p. 359 et seq.
§ XXIII.] JNTEGRABILITY OF MONGE'S EQUATIONS. 335
308. The auxiliary equations (5) are known as Mange's equa-
tions. The first is a quadratic for the ratio dy : dx, and is there-
fore decomposable into two equations of the form dy = mdx.
Employing either of these the second equation becomes a rela-
tion between dp, dq and dx or dy. These two equations, taken
in connection with
dz = pdx ■\- qdy,
form a system of three ordinary differential equations between
the five variables x, y, z, p and q. Since four equations are
needed to form a determinate system for five variables, it is
only when a certain condition is fulfilled that it is possible to
obtain by the combination of these three equations an exact
equation giving an integral 21 = a. Again, a second condition of
integrability * must be fulfilled in order that the second integral
V = b shall be possible. These two conditions are in fact the
same as those mentioned in Art. 303, as necessary to the exist-
ence of an intermediate integral containing an arbitrary function.
309. If R, S and T in the given equation contain x and y
only, the first of Monge's equations is integrable of itself. Given,
for example, the equation
X ■\-y
xr- {x+y)s+yt=-^^^{p-S) (i)
Monge's equations are
xdy^ -\- (x -\-y)dydx +ydx' = o, (2)
X -\- y
xdpdy -\- ydqdx = ^^-^{p - q)dydx (3)
* When there is a. deficiency of one equation in a system, a single condition
must be satisfied to make an integral possible, just as a single condition is necessary
when one equation is given between three variables. Supposing one integral found,
one of the variables can be completely eliminated; there is still a deficiency of one
equation in the reduced system, and again a condition must be fulfilled to make a
second integral possible.
336 EQUATIONS OF THE SECOND ORDER. [Art. 309.
Equation (2) may be written
{dy + dx) {xdy +ydx) = o.
Taking the second factor, we have
xdy + ydx = o,
which gives the integral
xy= a, (4)
and converts equation (3) into
dp — dq _dx — dy
p — q X — y
This gives for the second integral
p-q
X — y ^^-^
Hence we have for the intermediate integral
' ■^^zr-y = ^i=^y) (6)
To solve this equation of the first order, Lagrange's equations
are
dx = — dy = , (7)
{x-y)^{xyy ^^'
of which the first gives
x+y = a (8)
For the second integral we readily obtain from equations (7)
xdx+ydy = / ,
{xy)
whence
<^{xy)d{xy) = dz.
§ XXIII.] EXAMPLES OF MONGE'S METHOD. 337
Since <^ is arbitrary, the integral of the first member is an arbi-
trary function of xy, hence we may write
z-(xy) =P; (9)
and finally putting ji = i/'(a), we have
z = (l>{xy) + tl,{x+y), (10)
which is therefore the general integral of equation (i).
Another intermediate integral might have been found, but
less readily, by employing the other factor of equation (2).
310. When either of the variables z, p ox q is contained in
R, S or T, the first of Monge's equations is integrable only in
connection with dz — pdx + qdy. For example, given the equation
qif _ 2pqs + p'^t = o,
Monge's equations are
^dy^ + 2.pqdydx + p^dx^ = o,
and
q'dpdy + p'dqdx = o.
The first is a perfect square and gives only
qdy + pdx = o,
which converts the second into
qdp — pdq = o.
Hence the integrals
z = a, and / = iq,
and the intermediate integral
p = q(l>(z).
For this Lagi-ange's equations are
338 EQUATIONS OF THE SECOND ORDER. [Art. 3IO.
dx = ^-^ = — ;
(z) = \p{z).
In this example but one intermediate integral can be found ;
the form of the final equation is that considered in Art. 301.
311. In the following example, the second of Monge's equa-
tions must be combined with ds =pdx -f qdy. Given
r—i= 1 — : (i)
X +y' ^ '
for which Monge's equations are
dy^ — dx'^ =0, (2)
dpdy — dqdx -\- ^^j—dydx = o (3)
Taking from equation (2)
dy — dx ■= o,
whence the integral
y= X -\-a, (4)
equation (3) becomes
or
{^x + a) {dp — dq) + i^pdx = o (5)
To ascertain whether this is an exact equation, subtract from
the first member the differential of (2x -\- a) {p — q), which is
{■zx -\- a){dp — dq) + 2pdx — 2qdx.
§ XXIII.] EXAMPLES OF MONGE'S METHOD. 339
The remainder is
2/{y-x) (7)
Lagrange's equations now are
J.\- _ _ dy _ ,k
-v+;'~ x-^y~ ^{y — x) — 2Z
whence we have the integral
x+y=a, (8)
which converts the relation between dy and d:: into
dz 2; _ <^(-.l' — a)
d\' a a
The integral of this last equation is
zf~'^=-iy^4>{2y-a)Jy + p (9)
Finally using equation (8) and putting ^ = -"/-(a), we have
{x+y)ze~^^y = -[{2y-a)dy + ,p(^x+y), . (10)
where x + v is to be put for a after the indicated integration.
312. In this example it was not possible to obtain the second
integral required in Lagrange's process in a form containing a
simple arbitrarv function of the form ^(//), as was done in finding
equation (9), Art. 309. Thus the final integral in the present
I.
r=/(jc,j'),
2.
3-
t-q = e^ J^e^,
4-
p + r= xy,
5-
xr+p= xy,
6.
zr+p^= 7,xy',
7-
r+p-=y-.
8.
zs — Q/> = — ,
xy
9-
340 EQUATIONS OF THE SECOND ORDER. [Art. 312.
case is not of the form considered in Art. 300. In the case of
a primitive of the present kind, there is but one intermediate
integral. Accordingly, it will be found that, had we employed
the other factor of equation (2), the resulting system of Monge's
equations would not have been integrable.
Examples XXIII.
Solve the following partial differential equations : —
2 = I f{x,y)dx^ + x{y) + \j/{y)
z = ^x^ logy + axy + {x) + ^{y)
z=y{e^ — e^) + (x) + e^>l/{x)
z = ^x^y — xy+ {y) + e-^il/{y)
z = ix'y + 4>{y) log.» + il/{y)
& = x'^y + x(j>(y) + ^{y)
z = log {e^y^'iy) - e-^y^ + il^iy)
z = {x)tlj{y)x^°sy
z= (x +y) logy + ^(«) + i'(x +y)
10. ps — qr =■ o, X = <^(x) + i/'(^)
1 1 . x'f + 2xys + jv^^ =0, z = x<|>i-] + ^j/i-
l^. r — a'f = o, z = (l>(y + ax) + {(/(y — ax)
13. x^'r - ff =qy-px, z = ^ [A + ^{xy)
14. q{\ ->r q)r - {p + q -\- 2pq)s + p{i +p)t= o,
x= {z) 4- ,p(^x+y + z)
15. {b + cqYr -2{d + cq) (a + cp)s 4- (« + cpyt = o,
y 4- x^{ax + by + cz) = \l/{ax + by + cz)
§ XXIV.] LINEAR EQUATIONS. 34I
XXIV.
Linear Equations.
313. A partial differential equation which is linear with re-
spect to the independent variable z and its derivatives may be
written in the symbolic form
F{D,D')z=V, (i)
where
D=—, D' = —
dx dy
and F" is a function of x and y. We have occasion to consider
solutions only in the form
z=f{x,y),
and shall therefore speak of a value of z which satisfies equa-
tion (i) as an integral. Since the result of operating with
F(D, D') upon the sum of several functions of x and y is obvi-
ously the sum of the results of operating upon the functions
separately, the sum of a particular integral of equation (i) and
the most general integral of
F{D, D')z =0 (2)
will constitute the general integral of equation (i). Hence, as
in the case of ordinary differential equations, the general in-
tegral of equation (2) is called the complementary function for
equation (i).
So also, as in the case of ordinary differential equations, when
the second member is zero, the product of an integral and an
arbitrary constant is also an integral ; but this does not, as in
the former case, lead to a term of the general integral, since
342 HOMOGENEOUS LINEAR EQUATIONS. [Art. 3 I 3.
such a term should contain an arbitrary function. It is, in fact,
only in special cases that the general integral consists of sepa-
rate terms involving arbitrary functions.
Homogeneous Equations with Constant Coefficients.
314. The simplest case is that in which the equation is of
the form
■ Ao-—-\-A^- + . . ■ + A^—- = o, . . . (i)
dx" dx" -^dy dy"
the derivatives contained being all of the same order, and their
coefficients being constants. Let us assume
2 = i{y + m^x) + ^{y + m^) + . . . + ii>„{y + m„x), . (3)
where <^i, ^2, . . . , <^« are arbitrary functions.
Given, for example, the equation
' + 2ax).
315. Equation (i) of the preceding article, when written
symbolically, is
(^o^« + A^D»- 'Z>' + . . . + A„D''')z = o,
or, resolving into symbolic factors,
{D - m^D') {D - mM) . . . {D - tn„D')z = o. . . (4)
Since the factors are commutative, this equation is evidently
satisfied by the integrals of the several equations,
(Z> — m^£>')z = 0, (Z> — m^I)')z = 0, ... (Z» — m„Z>')z = o.
Accordingly the several terms of the general integral (3) are
the integrals of these separate equations.
Again, the equation may be written
(I)— (j + mx) as equivalent to the multiplier m,
thus
— <^(_y + mx) = m(.y+>»x) =/{m)^(y + mx) ;
344 LINEAR EQUATIONS. [Art. 315.
SO that equation (5) is satisfied by <^{y 4- mx) when f{m) =0,
whatever be the form of the function <^.
316. The solution of the component equations, of which the
form is
(Z)- mD')z=o (i)
may be symboHcally derived from that of the corresponding
case of ordinary differential equations. For, if we regard D' in
equation (i) as constant, its integral is
z = Ce'"0'x^
where C is a constant of integration. Replacing C by {y), as
usual in integrating with respect to one variable only, we have
for the symbolic solution
z = ^'^^'^Xj;), (2)
where <^(j/) is written after the symbol because D' operates
upon it, though it does not operate upon x. The symbol e""''^'
is to be interpreted exactly as if D' were an algebraic quantity.
Thus
"**«=('+"'l+Tf $ + •■>«
= 4>{y) + mx4><{y) +^^"(y) + ..,,
2 !
or
gmxD'^(^y-^ = ,^(j^ + mx),
by Taylor's theorem, of which this is in fact the symbolic state-
ment (Diff. Calc. Art. 176).
It should be noticed that the process of verifying the identity
{Z> — mZ>')e'"^o'(y + m,x),
Lagrange's equations are
^^ _ _ ^ - ^^
Wt (y + m^x) + i^(ji; + m^x) ;
and, regarding ^ and - m£>yz = o,
z = e'"^^'[_x{y) +>/'(7)],
that is,
z = x^{y + mx) +il>^{y + mx), (2)
but this is equivalent to the preceding result ; for we may write
it in the form
z = (y + mx — mx') <^i {y + mx^ + "/"i (j^ + ^x) ;
and, since (j + mx) z(j' + inx) + "Ai (j)' + tnx) and — m<^r{y + mx)
§ XXIV.] CASE OF IMAGINARY ROOTS. 347
are two independent arbitrary functions of y-\-mx, they may
be represented by ^ and ^, the equation thus becoming identical
with equation (i).
In like manner, if the equation /[ — ^, J = o has r equal roots,
the terms corresponding to {D — mD'Y are
x^~''4>^{y + mx) + x'—^<^^{y + mx) + . . . + 4>r{y + mx).
Case of Imaginary Roots.
319. When the equation has a pair of imaginary roots, fi ± iv,
the corresponding terms in the general integral are
z = ^(^ + /cta: + ivx) + \\i(^y + /jLX — ivx) ;
or, putting u=y + fix, v = vx,
<^(« + iv) + !/'(« — iv).
To reduce this expression to a real form, assume
/'i(« + iv) — il/i{u — iv)'].
In this expression <^i and fi are arbitrary functions, since <^ and
1/' were arbitrary ; but giving any real forms to <^i and i/-!, the two
terms are real functions of zt and v, that is to say, real functions
of ;ir and_y.
348 LINEAR EQUATIONS. [Art. 3 1 9.
Given, for example, the equation
d'^z , d'^z
1 = o,
dx^ dy
of whicii the solution in the general form is
z = ri.x - «»] +^-[}f/^{x + iy) - \p^{x - «»].
If, for instance, we assume ^i(^) = t^ and i/*! (t) = e*, we have the
particular solution in real form
z = ^3 — 2,xy'^ + e" sinj,
which is readily verified.
The Particular Integral.
320. The methods explained in the preceding articles enable
us to find the complementary function for an equation of the
form
F{D, D')z = V,
when F{D, D') is a homogeneous function of D and D', and Vz.
function of x and y. The particular integral, which is denoted
by
I V
F{D,D') '
can also in this case be readily found.
Resolving the homogeneous symboLi^(Z>, D') into factors, we
may write
F{D, D') = {D- m,D') {D - m^D') . . . {D - m„D'),
§ XXIV.] THE PARTICULAR INTEGRAL. 349
and the inverse symbol may be separated, as in Art.
105, into partial fractions of the form
where the numerators are numerical quantities, and r is unity
except when multiple roots occur. It is therefore only neces-
sary to interpret the symbol
{£>- mUy
321. For this purpose we employ the formula
^ (Z>) e<^V= e''^ {D + a) V,
proved in Art. 116. Putting mD' in place of a* this formula
gives
1 ^{x, y) = ^ e^^'^'e-""'^'*(^, y + mx) = result of putting y+mx for >< in (Z> + mD') i{x, y),
which expresses an obvious truth.
350
LINEAR EQUATIONS.
[Art. 321.
$(:«, y — mx)dx =
^{ly-m^)ai,
this may be expressed by the equation
D — mD
-,*(*, jO = ^{i, y + mx — m^) dt
(2)
In Uke manner, for the terms corresponding to multiple roots
oif{ni) = o, we have
322. There are certain methods by which, in the case of
special forms of the operand, the result may be obtained more
expeditiously than by the general method just given. Some of
these, which apply as well when the equation is not homogene-
ous, will be found in Arts. 328-334. The following applies only
when the equation is homogeneous.
Suppose the second inembe}' to be of the form ^(ax + by). The
equation may be written in the form
F{D, D')z = J>fi^ 2 = ^{ax + by).
It is readily seen that
/("-VC^* + by) =/Q ^{ax + by).
We have, therefore, for the particular integral
z=—^j^{ax-^by)=-LA[...U{ax + by)dx^, . (i)
^' + DD' — 2Z)'==
= ^ \ ■imidt'' = - sini'= \ sin {x + 2v).
I + 2 - 8jJ 5 5 ^
Adding the complementary function,
z = ,^{y-\- x') + ^{y — 2:«) + i sin (x + zj^).
323. When F(a, b) = o, the operand is of the form of one of
the terms of the complementary function. The method then
fails, the expression given in the preceding article representing
a term included in the complementary function, with an infin-
ite coefficient. In this case, after applying the method to all
the factors of the operative symbol, except that which vanishes
when we put D = a and U= b, the solution may be completed
by means of the formula
-,f{y + '«■») = ^f^y + ^'•*) » -
D'- mD''
which results immediately from equation (i). Art. 321.
* This integral involves an expression of the form Ai"~'^ + Bt"-'' + . . . + Z.
in which A, B, . . .,L are arbitrary constants, but such an expression is included in
the complementary function. It must be remembered that the multiple integral in
equation (l) is not to be regarded as involving an arbitrary function oiy.
352 LINEAR EQUATIONS. [Art. 323.
Thus, if in the example given in the preceding article the
second member had beeny(;t:+^), we should have had
2 = fix + v)
= — J — e^D- (-"'''/{x + r)
= _L e^o'l-/(y) = — e^^'x/(y)
= ^xf(x + y)dx.
The Non-Homogeneous Equation.
324. When the equation
F{n,D')z = o (i)
is not homogeneous with respect to D and D', the solution can-
not generally be expressed in a form involving arbitrary func-
tions. Let us, however, assume
z = ce''^-^>'y, (2)
where c, h and k arc constants. Substituting in equation (i),
we have, since Z'^^^+*J' = /2^*^+*J' and D'e'^'+^y = ke'^'-^^y,
cF{h, k)e''^ + ''y = o.
Thus we have a solution of the assumed form, if h and k satisfy
the relation
J'ih, k) = o, (3)
c being arbitrary. Let equation (3) be solved for h in terms of
k. Now if F{h, k) is homogeneous, we shall have roots of the
form
k = m^k, h = mji, . . . , h-= m„k ;
§ XXIV.] THE NON-HOMOGENEOUS EQUATION. 353
and, since the sum of any number of terms of the form (2) which
satisfy the condition (3) is also a solution, the equation will be
satisfied by any expression of the form
where m has any one of the values m^, m^, . . ., m„. But, since
for a given value of m this expression is a series of powers of
gy+mx ■v(rith arbitrary coeificients and exponents, it is equivalent
to an arbitrary function of ^^+»'^, that is to say, it denotes an
arbitrary function of ^ + mx. This agrees with the result other-
wise found in Art. 314.
325. Again, if F{p, D') can be resolved into factors, and
one of these is of the form D — mD' — d, so that F(/i, /^) = o is
satisfied by
h = tnk + b,
equation (i) will be satisfied by an expression of^the form
z = '%ce'''^y + '«^) + '"',
where m. and b are fixed and c and k are arbitrary. But this ex-
pression is equivalent to the product of e''^ into an arbitrary
function oi y + mx. Thus, corresponding to every factor of the
form D — mD' — ^ we have a solution of the form
z = ^*^(^ {y -f- mx) .
Given, for example, the equation
d^z d'z , dz dz
1 1 = 0,
dx" dy^ dx dy
or
{D + 2?') (i? - Z>' + i)z = o ;
the general integral is
z= ,^{y — X) + f -^i/^Cj' + .ar).
354 LINEAR EQUATIONS. [Art. 325.
We might also have found the solution in the fori
but, writing the last term in the form ey*^-''\^-^{y + x), this
agrees with the previous result if ^^+/Wy satisfies equation
dh'' dhr
(i) ; thus we have the series of integrals
ce^^+fWy [x+f{h)y]
ce>'^+/Wy\lx +f{h)yy +f\h)y\
cek-+/Wy\ [^ +f{h)yY + zf\h)y \_x +/'(A)7] +f"{h)y\
(5)
For example, in the case of the equation (Z?^ — D')z = o, the
integral (?^^+*'^ gives rise to the integrals
«*^+*'^ {x+zhy),
e^^ + '''yi{x + 2hyy+ zy'],
e^^+'^'^Kx + 2hyy + (>y{x + 2hy)\
^x+h^y^(^y. j^ 2hy)^ + i2y{x + 2hyy + i2_>'=],
In particular, putting h = o, we have the algebraic integral
z=z c-,x + (^(x' + 2y) + Cj{x3 + 6xy)
+ Ci{x* + 'i2X''y + I2J1'=') + . . .
Special Methods for the Particular Integral.
328. The particular integral of the equation
F{D, I)')z = V^
356 LINEAR EQUATIONS. [Art. 32S.
is readily fotiiid in the case of certain special forms of the func-
tion V.
In the first place, suppose V to be of the forfn ^"^ +*■>'. Since
jj^ax + iy^^gax + hy ^nd D' e'^'^+^y = be"^ ^^^ , and F{D, D') consists
c f terms of the form D''D'^, we have
or
F{D, D')e'^ + iy =F{a, 3)6"^ + ^,
F{a, b^c^ + iy = e'^ + ^'y,
F{D, D')
where F{a, b) is a constant. Hence, except when F{a, b) = o,
we have
_ gax + ^^ ;^ gax + by ^
F{D, D') F{a, b)
Thus, when the operand is of the form e'^^'^''y, we may put a for
D and b for D\ except when the result introduces an infinite co-
efficient. Given, for example, the equation
{I> —D')z = e^^+y,
the particular integral is
Z = ^ e^x+y _; \(^x+y_
I>—D' '
329. In the exceptional case when F{a, b) — o, we may pro-
ceed as in Art. no. Thus, first changing a in the operand to
a + h, we ha:ve
I .^x+&x+iy— 1 e''''+iyfi^^x+-^-\-.
F(£>,jD') Fia+A,b) \
The first term of this development is included in the comple-
mentary function. Omitting it, we may therefore write for the
particular integral
§ XXIV.] SECOND MEMBER OF THE FORM e'-' + h. 357
{x + ^kx^ + . . .)e'^ + iJ',
F{a + h,b)
in which the coefficient takes the indeterminate form when k=o,
because F{a,b)=o, and its value is , where Fa{a, b)
Fa' {a, b)
denotes the derivative of F{a, b) with respect to a. Hence,
except when Fa {a, b) = o, we have
Fa' {a, b) ^ ^
In like manner, if Fa (a, b) = o, the second term of the de-
velopment is in the complementary function, and we proceed to
the third term. It is evident that we might also have obtained
the particular integral when F{a, (^) = o in the form
y
Fi'{a, b)
but the two results agree, for their difference,
e^^ + h; (2)
y
lFa'{a, b) Fi'{a, b)_
gax + by
is readily seen to be included in the first of the special forms
(5) of Art. 327, since a and b are admissible values of the h and
k of that article.
330. In the next place, let V be of the form sin {ax + by) or
cos {ax -\- by). We may proceed as in Arts, in and 112, and it
is to be noticed that we have, for these forms of the operand, not
only Z?" = — fl^ and Z''= = — b'^, but also DD' = — ab. Given, for
example, the equation
1 1 z = sm{x + 2y),
dx'^ dxdy dy
358 LINEAR EQUATIONS. [Art. 33O.
the particular integral is
sin {x + 2y) = — ; sin (x + 2y)
= ,^ _ sin {x + 27) = — jL [cos (x + 2y) + 2 sin(* + 2:^)].
Adding the complementary function, we have
z — e'^{y — x) + e-'^ijj{y) — J-j- cos {x + 2jc) — 3- sin {x + 2y) .
The anomalous case in which an infinite coefficient arises
may be treated like the corresponding case in ordinary differen-
tial equations.
331. Again let V be of the form x^y^, where r and s are
positive integers. In this case, we develop the inverse symbol
in ascending powers of D and D\ Thus, if the second member
in the example of the preceding article had contained the term
xy, the corresponding part of the particular integral would have
been found as follows :
^^~ I - {2> + DD' + D'f^^
= - [i + (i?- + DD' + £)')+{D' + DD' + D'Y + . . .]xy
= —\_i+D^ + DD' +D' + 2D'D''\ x'y
= — x^y — 2y — 2x — x^ — 4.
It will be noticed that, on account of the form of the operand, it
is unnecessary to retain in the development any terms containing
higher powers than Z>= and D'. Again, had the operand been
xy, we might have rejected i?^ in the denominator thus :
T I
-xy= — - — -— — xy
D'+DD'+D'-i i-D'{i+D)
= -[i+7?'(i +D)^xy= -xy-x — i.
§ XXIV.] SECOND MEMBEk OF THE FORM x'-f. 359
332- When the symbol F{D, D') contains no absolute term,
we expand the inverse symbol ,in ascending powers of either
D or D\ first dividing the denominator by the term containing
the lowest power of the selected symbol. For example, given
the equation
dx^ dxdy
for the particular integral we have to evaluate
In this case, it is best to develop in ascending powers of D\
because, with the given operand, a higher power of D than of
D' would have to be retained. Thus
I ..,.. I /. , D
xy = — I I + "5 — I x'y
= — x^'y +^3- x^ = ^ -\ .
JD^ D^ 12 20
Adding the complementary function,
z = <^(j)/) + ^{y + 7,x) + -^x^y + ^^5.
If we develop the symbol in ascending powers of D, the par-
ticular integral found will be
_ .»y ^y xy^
" 18 54 324'
The difference between the two particular integrals will be
found to be
which is included in the complementary function.
360 LINEAR EQUATIONS. [Art. 333.
333. Finally, when the operand is of the form e^'-^^yV, we
may employ the formula of reduction
F{D, D')e'^+hV= e<'^ + hF{I) + a, D' + b) V,
which is simply a double application of the formula of Art. 116.
For example,
J> - D' D^+ 2aD - n
— pax +a'^y _
2aD , D
2a
gax + a'-y I / J)
I ]X
^e'" + '''yf— -—\
\4« 4«7
2a D\ 2a
If we develop in powers of £>', we shall find
xe'"^+''''y = — e'"'+'''y(xv + av').
D^-D' \ J'-^ J' )
The difference between the two results is accounted for by the
special forms given in Art. 327 for the complementary function
in this example.
334. As another application of the formula, let us solve the
equation
•T- + ■^— r — 6 — - = Jf^ sin {x -\-y).
dx^ dxdy dy^
The particular integral is
z = the coefficient of « in — — — — - e'-^+'yx''.
J> +£>£)'- 61)''
Now
_ gix + iy^i = e'^ + '> : ^ IP
]> + DD' - dD'^ (Z* + ^ + i{D + i)- 6i'
f I i^Z
. — £ix + i'y _
§ XXIV.] SECOND MEMBER OF THE FORM e'^ + h'V. 361
and by development we find
2 — _ _ y^ _ £3
therefore
— -, e"+'yx' = [cos(a: +y)+ism(x -\-y)'\\— -^^-'^ .
Taking the coefficient of i, and adding the complementary
function,
- = f — - ^ j sin(a; Jt-y)-^ cos(:c +>-) + ^(>' + 2a-) + ^{y - 3-*) •
Linear Equations with Variable Coefficients.
335. In some eases a linear equation with variable coeffi-
cients can be reduced, by a change of the independent variables,
to a form in which the coefficients are constant. As an illus-
tration, let us take the equation
I d^z \ dz I dH \ dz , ,
3C^ dx^ x^ dx f dy^ y^ dy
The first member may be written in the form
I fi d^z I
X \_x dx^ x'^
dx
/I d_ I dz
x dx X dx
Hence, if we put xdx = di, whence i = ^jr", and in like manner
■q = ^j'', the equation becomes
d^^d^ ,.
dt drf ^ '
The integral of this equation is s={i+ri)-'t-\li{^— rj); hence
that of equation (i) is
z = <^{x^ +y) + ^{x^ -y^).
362 LINEAR EQUATIONS. [Art. 336,
336. In particular, it is to be noticed' that an equation all of
whose terms are of the form
is reducible to the form with constant coefficients, like the cor-
responding case in ordinary equations, Art. 123, by the trans-
formations i = log ;r, 17 = logy, which give
d _ d d _ d
dx di, dy dr\
But, if we put * = X—- and i?' = y -— , we may still regard x and y
dx dy
as the independent variables ; the transformation is then effected
by the formula
x'^f-^^^=^x^D^.fD''=d-{f^-\)...{^^-r^\)d'(d>-\)...{d'-s->r\\
and the equation reduced to the form
F{fi, ^')z = V.
The solution of this equation may therefore be derived from
that of the equation F{D, D') z = V,hy replacing x and y by
logjp and logy; or it may, as in the following articles, be ob-
tained directly by processes similar to those employed in deriv-
ing the solution of F{D, D')z = V.
337. Since
^x^y'' — rxTf, d>x''y^ = sx''y,
it is obvious that
F{&, ^')x'-y == F{r, s)x'-y'' (i)
§ XXIV.] THE EQUATION F{fi;-d')z = o. 363
Hence, if in
F{fi,^V)z=o, (2)
we assume
z = cx^f,
the result is
cF{r, s)x'^y^ = o,
and we have a solution of the proposed form if F(r, s) = o.
Hence the general solution of equation (2) is
z = -s,cx'-r, (3)
where
F{r, s) = o, (4)
that is, ^ is a series in which the coefficients are arbitrary,
and the exponents of x and jy are connected by the single
relation (4).
Now let equation (4) be solved for r in terms oi s ; if the
function F{^9, 1?') be homogeneous in ^ and ^^', the equation will
have roots of the form
r = m^s, r = m^s, etc.,
and to each root will correspond a solution of the form
y = ■2,t:{yx'"y.
I
But this represents an arbitrary function of j/x'"- Thus to each
factor of i^('9, &') of the form
i9 — m»',
there corresponds an independent term of the form
z = tf>(yx'")
in the solution of equation (2).
364 LINEAR EQUATIONS. [Art. 33/,
Again, corresponding to a factor of the form
we have the root r = ?«j + b, for Fi^, s) = o; and hence the solu-
tion z = S(:(j-«''")^-«'*, or
z = x^(j>{yx'").
338. For the particular integral of the equation
we may suppose V to be expanded in products of powers of ;i:
and,?/. By equation (i) of the preceding article, we have
— x" r = x" V*,
F{&,&') ^ F{a,b) ^'
which gives the particular integral, except when F{a, b) = o.
When this is the case, we have, first putting a + ^ in place of a,
1 Xa+Ziyi — 1 x'y^ii + h log^ + • • ■)>
F{^,^V) ^ F{a + h,b) ^ ^ ^ "
or, rejecting the first term of the expansion, which is included in
the complementary function, and then putting h = o,
- x^y^ = x''y^ log X.
F{^% *') -^ Fa\a, b)
339. As an illustration, let us take the equation
;y= y^ — = xy,
dx' ^ dy=- ^'
which, when reduced to the ^?-form, is
^'^{•'^ - i)z — fi'{ft' — i)z = xy,
§ XXIV.] THE EQUATION F(JS;^')z=V. 365
or
The complementary function is
for the particular integral,
II II
xy = ■ xy
1> — .>' t? + .?' — I -^ I + I — I t> — t?'
'^^^^^T^'*^'"^*-^'^^'*'-^^^ +>^log^ + ..•).
Of, rejecting the term —xy included in <^(^xy), and putting k = o,
h
z = (■*y) +■*>/'(-) + ■^jc log X,
340. The symbol '? + '^' may be particularly mentioned on
account of its relation to the homogeneous function of x and y.
Putting
we have vx^y' = {r+s)x^y' ; hence, if ?/„ denotes a homogeneous
function of x and 7 of the «th degree, we have
■7rU„ = nu„,
where u„ is not necessarily an algebraic function, but may be any
function of the form x"/(^\ This is, in fact, the first of Euler's
theorem concerning homogeneous functions. See Diff. Calc,
Art. 412.
As an example of an equation expressible by means of the
single symbol w, let us take
366 LINEAR EQUATIONS. [Art. 34O.
dx" doc"~-^dy 2 dx^-^dy^
The first member can be shown to be equivalent to
7r(7r — l) . . . (ir — « + \)z.
Denoting this by F{Tt)z, we have
F{ir)u^ = m{m~ 1) . . . {m — n+ ■!.)u^, . . . (2)
which, when FItt) is expressed as in equation (i), is the general
case of Euler's theorem. Thus the complementary function for
equation (i) is
«0 + «I + «2 + . . . + «»-I.
Let V contain the given homogeneous function H^, equation (2)
gives for the corresponding term in the particular integral
■Hmi
m{m — i) . . . (m — « + i)
except when m is an integer less than n. In this case F(^) will
contain the factor tr — m, and putting F{7r) = (ir — m){Tr) we
readily obtain as in Art. 338
Examples XXIV.
Solve the following partial differential equations : —
d^z d^'z d'z . , \ , , / , \
d^z d^z , d^z I
2. 2 1 = — ,
dx'dy dxdy^ dy^ x^
z = <^(.r) + iIj{x -\-y) + xx{x + r) — _y log.r.
§ XXIV.] EXAMPLES. 367
d'^z , d^z , , d^z I
3- :rz + S-j-T+^
dx^ dxdy dy y — 2x'
z = <^(jc — 2x) + (y + «) + e-'^ij/iy + 2x).
^ d^z d^z dz , dz ,,„ ,
^ dx^ dy^ ^ dx^ ^ dy ■^'
z=^{x +y) + e^^\ii (jy — x)
- ye'*'^ - {-hx^ + \x^y + \x^ + \xy + -i^x).
6. J^ + a— +b— + abz= e^y^'^,
d}cdy dx dy
z = e-''y(y) + e-''\j/{x +y) + ^ sin (a: + 2y) — xe".
o d'z d'z , dz dz ■ / , \
8. 1 = 2sm{x+y),
dx' dxdy dx dy
z = e-''4>(y) + "AC^ +y) + ^ [.sm{x +y) — cos (x + j)].
g. ^ -a^ = ^'"^cos;/v,
^ dx dy
= '
.■*/ \^J n{n — i)
^d'^z , d'z , ^d'z dz dz ,
13. X'- — \-2xy——-+y^- nx ny i- nz = o,
dx' dxdy dy' dx dy
' = ^''4i)+^i'{^y
d^z dz f
''^' ''''^^'^^^y~''Yx^°' z=^{x+yY{x)dx + ,i,{y).
^ dx^ dy ^ ^■^''
z = ^ce^'y--^'^ + ey-'{ix' + ^x- iy + ly).
16. Derive the particular integral of
<&3 dy ^ ^■^'
in the form z = xye^ •*'.
368 LINEAR EQUATIONS. [Art. 34O.
II. mn {iri^ -\- n?-) \-m.n \- mn^ m^n —
dx'' dxdy dy^ dx dy
= cos {kx + /)') + cos {mx + ny^,
2 = ^ (wy + mx^ + e-"^\\i{my + «jc)
. »^«sin(,^jc + /)<) — (»z^ — «/) cos(^^ + /f)
(nk — ml) [m^n^ + {mk — «/)=]
■ w;zjc co's,{mx + %y) + (^' — n'')x sin{mx + ^y)
^\xy \xj n{n — i)
^d'z , d^'z , ^d^'z dz dz ,
13. x''-— + 2xy—-—-+y^- nx- ny—- + nz = o,
dx^ dxdy dy' dx dy
d^z dz (
14. (''+y')-^-'^-^ = °' z=j(x+yy{x)dx + ^l,(y).
z = ^c^'y-"'- + eyHx' + ix- iy^ + iy) .
16. Derive the particular integral of
d^z , d'z
dx" dy'
in the form z = xyt" " .
.rr, witf jfr.iafe. iL