1 — n cot
that is, the orthogonal trajectories are
6 -6' = a.
[In general, for a bipolar equation
f(r,r') = A,
the differential equation of the orthogonal trajectories is
Mdd-r'^de'=Q,
or or
with the relations r sin 8 = r sin 6', r cos 6 + r' cos & = 2c]
48 CHAPTER IV. MISCELLANEOUS EXAMPLES
Ex. 33. The expression of the property is
fx fx
x \ y 2 dx = 2y xydx,
J n. J a
leading to the relation
d^ + *dy_4/dyy =Q _
dx* x dx y \dxj
The primitive (by § 67, Ex. 2) is
A B .
of y i
substituting the value of y in the original relation, we find A = a?.
Hence the result.
Ex. 34. For any curve rolling on the axis of as,
r = y(l + f)i,
where y is the perpendicular from the pole on the tangent, and
t = dyjdx. Also
A_A A/^V-
hence the equation, for the particular curve, is
1— m
y = a (I + t 2 ) 1 *™ .
When m = -|, the integral equation is
y = a cosh {(x — A)ja],
a catenary.
When m = 2, the integral equation is
x - A = a 2 /(?/ 4 - a 4 )" - dy,
the elastica curve, the finite expression of which depends upon
elliptic functions*.
* Halphen, Fonctions elliptiques, vol. ii, chap. v.
CHAPTER V.
d?
§ 83. Ex. 2. (i) The equation is -^ (xy) + &x . xy = ; so p. 150
taking the solution of the preceding question with ex instead of
x, we have
xy = A\l-^ (exf + ^j (cxy - ^^ (exf + . .
D f 2 . ., 2.5..,. 2 . o . 8 . ., i
+ I ¥1 ^ + IT (ca3) ~ ~JoF (ca,) + " j '
.... . f, ax* a?x s a?x u
(xx) 2/ = ^ (1 -3^+3^7778 ~ 3. 4.7 ...11.12 + -
f a^c 4 a 2 « 8 a 3 « 12 )
+ j 4TT5 + 4.5.8.9~4.5.8.9.12.l:3 + """J '
Ex. 3. Let y = A when « = ; the equation gives -r^-= — mA ; p. 151
and as d n+1 y d n i/ d n ~ x y _
X dx^ x + n dx^ + m dx^- 0>
. d n y , ., . „ m n .
we have -y-^ = (- 1)" — r A.
dx n K ' n\
Hence the required integral.
§ 85. Ex. 1. The equation can be written p. 153
x i{ x Tx- l+2n ) y+m ^ =0 -
The values of m, are given by m, (m 1 — 1 + 2w) = ; and, with the
notation of the text,
Af.m^ (m M — 1 + 2m) + mA^ = 0.
Thus -4.JA+1 = for each value of X.
For m 1 = 0, the value of y is
m „ »n 2
2(2m + l) .2.4(2n + l)(2n + 3)
For 7«! = 1 — 2??, the value of y is
f 771 lYv
B x*-™ jl - 2 (3 _ 2b) « 2 + 2 .4(3-2n)(5-2n) ** ~ '
f. 4
50 CHAPTER V. § 85
p. 155 Ex. 3. Using the result of § 83, Ex. 3, and taking m = 1, a
particular solution is
— tA/ iAj \Aj
= 1 1 u
111! 2!2! 3! 3!
So, after the explanations in Ex. 2 (p. 154), we take
y = Y(A + B log x) + w,
and find w = 2B j«_ — ^ + _ a? ~^«* + ••
[In w, the coefficient of # n is
« n (-l)" A 11 1\
(?z!) 2 V. 2 d nj
The equation should also be solved by the method of Frobenius,
Chap, vi, Note i.]
Ex. 4.
... . I a 2 , a 2 (a 2 -2 2 K a 2 (a 2 - 2 2 ) (a 2 - 4 2 ) „ )
(!) y = ^{i-2i^-+ 4, ' «?-- g! x — ••}
, nJ a2 - p ™3 , (« 2 -l 2 )(a 2 -3 2 ) ,
73
= J. cos (a sin -1 ;*;) + — sin (a sin _1 «) ;
a x '
(ii) Change the variable from x to £, by the relation x 2 = z;
the equation becomes
*(l-*)y" + (l-2*)y'-iy = 0,
and a particular solution is
*-' + (i)'-(o)"---
which can be expressed in the form - (1 - z sin 2 6Y^d9.
tt Jo
The new equation is unaltered if z be changed to / by the
relation z = 1 — z. So another particular solution is
y-.
-+GMH)"—-
2 r^
which can be expressed in the form — (1 — z' sin 2 8)~^dd.
71" Jo
The primitive is y = Ay l + By 2 .
[The quantities y 1 and y 2 , when multiplied by ^tt, are the
quarter-periods in the Jacobian elliptic functions.]
chapter v. §§ 86, 87, 90 51
§ 86. Ex. I. The primitive is p. 156
yx~ x e- x = A + BJx~*er x dcc.
Ex. 2. The primitive is
x — |# 2 J x (x — -^a; 2 ) 2
§ 87. Ex. 2. The primitive is p. 158
+ fe2 { l ~ 3^4* + (8 , _ 4)'^ _ 4 )* ' + -} •
Ex. 3. The primitive is
X y e -mx = A (I - ^qx) + B (1 + \qx) e~i x .
§ 90. Ex. 1. The result can be obtained by actually carrying p. 162
out the n differentiations of the expansion of (* 2 — 1)" in powers
of x — an unsuggestive method.
The result can be established by the use of one of Lagrange's
expansions (in books on the differential calculus, it is usually called
Lagrange's theorem) in connection with the equation
y = x + \z (y- - 1),
so as to expand y in powers of z. We take the root y given by
1 1
2
y = (1-2x2 + 2'-)
J 2 Z
The theorem, just mentioned, gives
±-±(l-2xz + zrf = x+2 _ a — , -f^j {(x>- 1)"}.
z z v „ = i 2 n .n\ dx n 1 lv '
Differentiation with respect to * leads at once to the relation
1 go 7 n fin
(1 - 2XZ + Z*)~t = 2 s/ ; , |- {(tf 2 - If}.
Expand (1 - 2x2 + z 2 ) * in ascending powers of z, in the form
l+£(2a*-*«)+ ~ (2fl*-*)»+...;
on selecting the coefficient of 2" in this expansion, it is found to
be the quantity denoted by P n in the text.
4—2
52 CHAPTER V. § 90
Ex. 2. The first part has been proved in the preceding example.
For the second part, we have v = %z n P n ; so
= -2,n(n+l)z n P a
d 2 (zv)
= — z — - — - .
dz 2
[The second part can be established also as follows. Let xi
denote (1 - 2xz + z 2 )? ; then
du z du z — x
dx u' dz u
d 2 u_
dx 2 ~
z 2 o 2 u 1 (z- x) 2 1-x 2
u*' dz 2 u v? v?
so that (l-a?)=-, + z 3 -r— = 0.
ox- dz 2
Co„se q „e„« y , £|(l--»g} + *£(£)-ft
or, using the relations — = = — zv, we have
ox u
Ex. 3. By Rolle's theorem*, an uneven number of roots of
f'(x) = must lie between any two consecutive real roots of a
polynomial equation f(x)= ; and an r-ple root of f(x) = is an
(r — l)ple root oif'(x) = 0. The roots of (a; 2 — l)' 1 = are 1, repeated
n times, and — 1, repeated n times. Hence the roots of -=~ {(x 2 — l) n }
are 1, repeated n — 1 times ; — 1 repeated n — 1 times ; and a root
d 2
(it is zero) between 1 and — 1. The roots of -j— 2 {{a? — \) n ) therefore
are 1, repeated n — 2 times; — 1 repeated n — 2 times; a root be-
tween 1 and 0, and a root between — 1 and 0, the two latter being-
equal in value and opposite in sign. And so on, in succession ;
all the n roots of P n are real ; each is numerically less than unity.
* Burnside and Panton's Theory of Equations, vol. i (7th ed.), § 71.
CHAPTER V. § 90 53
Ex. 4. In Ex. 2 it was proved that
(1 - 2zx + z*y% = 1 z n P n (x) ;
1 °°
take a> = 1 ; then = 1 z n P n (\),
i- — z m=0
so that P«(l) = 1, proving the proposition.
Ex. 5. (i) Differentiating the relation
(1 - 2zx + a*) - * = 2 «"P m (as)
m =
with respect to z, we have
(1 - 2zx + z i )~i{x -z)= 2 mz m ~ 1 P m (x).
m =
Multiply by 1 — 2zx + z 2 , and equate coefficients of z ll ~ l on the two
sides; we have
xP n ^ - P n _ 2 = nP n - (2ra - 1) ap,^ + (ft - 2) P n _ 2 ,
leading to the result ;
(ii) Multiply the differentiated relation in (iv) below by
x — z and equate to the differentiated expression in (i); then
X ~dx~ dx-= mPm -
Now ^J(i_ fl * ) ^l = _ n(w + i ) p B ,
= (n + l)P n ,
Integrate ; determine the constant of integration by the property
then, when x = l, P m is unity. The result follows.
(iii) This relation effectively is the same as (i).
(iv) Differentiating the relation in (i) with respect to x, we
have (after division by z)
{l-2zx.+ z*)* = 2 J""" 1 ^- 5 .
m=l ">&
From the preceding differentiation in (i), we have
1 - z* - (1 - 2xz + z*)
(1 - 2zx + z*f
= ~Z2mz m P m (x),
54 CHAPTER V. § 90
and therefore, taking the coefficients of z n ~ x ,
dP dP
~dx dx~ ~ K ~ ' n ' 1 '
Ex. 6. We have
C" d6 _ f" d0 it
Jo z — ip cos J o z + ip cos / z % + „2\i '
where p is real and i denotes (— 1)* Take
z = \ — ax, p= a (1 — a; 2 )^ j
then t = 1 ;
•'o l-a{x±(x i -\f - l)^cos 0}" d0.
J o
The second result follows from the substitution
[x ±{x i - 1)* cos 0} {x ± (x°- - 1)* cos Qn(x)
dx n
= (-1)
„ f] 2»n(n)n(«)
II(2n+l)
II(2w+l) 1 Yl(2n + 2) 1
~~njnj~~ x in+i + 2! n(n) x m+i + " '
= -(-2)»n(w)
^n + ( n + 1 )^+i + 2 - l ( n + 1 )( n + 2 )^
: +
(-2)«no) .
(* 2 -l)" +1 '
(ii), (iii) Substitute the foregoing expression in (i) for Q n
and compare coefficients.
Ex. 3. (a) Integrate the relation in Ex. 2, (iii); we have
n Q n+1 + (n + 1) Q„_! = (2n + 1) I « Q„ 1, so that of course x < y.
Then
1 1 x x*
u = =-+- + -+...,
y-x y y 2 y s
= y<> + yiPi{%) + 2/2-P2 (x) + ... ,
when each power of x is expressed in terms of the Legendre
functions; and the coefficients y , y lt y 2 , ... are aggregates of
negative powers of y. Now
d_ \ n j. du) _ 2 - 2xy _ d_ ( . 2 . du\
dx\ {L x) dx\~(y-xy~dy\ {i y) dyy
and therefore, when the expansion for u is substituted, and co-
efficients of P n (x) are compared, we have
|{ (W2) t) +)l(W + 1) ^ = °-
Thus y n is a solution of the Legendre equation ; it contains only
negative powers of y ; and therefore it can only be a multiple of
Qn(y), say
y„ = A n Q n (y).
Now take the coefficient of x n /y n+1 in the two forms of u. In
the second form, it cannot arise out of terms after Q n (y) P n (x),
because they do not contain ?/-< n+1 > ; and it cannot arise out of
earlier terms, because they do not contain x n . In A n Q n (y) P n (x),
the coefficient is
2?).! 2 n .nl nl
"2».n!m!(2n+l)!'
CHAPTER V. §§ 99, 103 57
hence A n = 2n + ~\,
and therefore -^— = I (2n + l)P n (x) Q n (y).
y — x B= o
§ 103. Ex. 1. From the results established in § 103, p. 180
~dx~ =Jn - 1 ~x n
2
= - j \nJ n - (n + 2) J n+2 + (n + 4) J n+i -...}.
Ex. 2. Verify at once by picking out the coefficient of z n from
the product of
x
xV z % (xV z 3
2) 2l + V2/ 3~!
l+2*+i'-.i ^H l:>) .",, '■
. -.-- H-.f..-g)'S-....
Also J,,, is the coefficient of (— l) n z~ n in the same expansion.
The relations in § 103 give
(°0> ~T- = — Ji) (b), 2 —t-^ = J n -i — Jn+i'> (c)i ~J™~ Jn+i + ,
cos (ic cos ) = J",, — 2./ 2 cos 2j> + 2J 4 cos 4 -
1 f"'
and therefore J" = — cos (x cos ) rfrf>.
7T 'o
f 00 - 1 /"" /*"■
Hence e~ ax J (bx) dx = — I e~ ax cos (bx cos <£) dxdd>
Jo ""Jo Jo
-f'-
7rJn « 2
6 2 COS 2 <£
= (a 2 + & 2 )~ i
p. 184 § 105. Ex. 1. See Hankel, i¥a^. 4tm?., t. i (1869), pp. 469-472.
ifo. 2. See Lommel, Studien ibber die Bessel'schen Functionen,
§§16, 17; Forsyth, Theory of Differential Equations, vol. iv, pp. 330
-1. See also a later part of the Treatise, containing the examples
solved in this volume ; pp. 253-6.
p. 186 § 106. Ex. We have (as in § 106)
y a J n _ 7- d 1 n _ A
n dx n dx x
The constant A may be determined from the lowest power of x,
which in J n is „ _, . . x n and in Y n , from the expression on p. 184, is
2" 11 (n) r
Substituting and comparing the coefficients of -, we have A = — 1:
thus j- dY n y- dJ n _ 1
* dx n dx x :
the required relation.
chapter v. §§ 107, 109, 110, 112 59
§ 107. Ex. The analysis shews that, if the assumption of p. 187
proceeding to the limit be justified or justifiable, the quantity in
the text satisfies Bessel's equation of order m and therefore has
the form
■AJ m + -oi m .
But the expression does not contain a logarithm, while part of the
expression for Y m is J m log x ; hence we must have B=0. We
infer the result, which analytically is not important except as in-
dicating a limiting relation between Legendre functions and Bessel
functions.
§ 109. Ex. The equation can be written p. 191
-r- + bu 2 = cx m ~ k ,
dz
where x h dx = dz ; the result follows from the theorem in the text.
§110. Ex.2, (i) A particular solutionis y = x 2 ; the primitive p. 194
(ii) With the notation of the text, P=2x, Q- — a?— 1,
R = 1 ; so the substitution is y = -j- , and the equation for u is
dx'
d 2 u du d , „ . A
dx 2 dx dx
the primitive of this equation is we* = A + BJe^dx, where
(iii) A particular solution is y = x + - ; proceed as in the
, . . 11
text, by substituting y = x + - + - ;
(iv) A particular solution is y=sirxx; proceed as in the
text, by substituting y = ~+smx.
§ 112. Ex. 1. Take m = 2, and n V^l in place of w; the primi- p. 199
tive (after the result in the text) is
y- x i^xl{^ sin(nx+B,) }'
which can be identified with the given result.
60 CHAPTER V. § 112
Ex. 2. Take qx = zi; the equation becomes
d 2 v i dv s _
cfo 2 ^ x dx
Next, take » = y**' ; the equation for y is
d« :
e-n«y =*(» + J)
ar
all the relations follow from the general results in the text.
[The memoir by Glaisher, containing these and other results,
should be consulted, especially §§ 28, 29 ; the reference is given,
p. 197 of the text.]
p. 200 Ex. 3. Substitute u = yx^ +1 ; then
d"y 2 . , . dy
The result follows from the general formulae in § 112.
Miscellaneous Examples at the end of Chapter V.
Ex. 1. (i) Change the variable from x to z by the relation
z s = x ; the equation becomes
dz* zdz yc ^- U >
the integral of which (§ 85, Ex. 1) is
f (Sczf ( Scsy (Bczf
y \ 1-2 1.2.4 1.2.3.4.6
I JL-- fl I (3cg)> I (3C " )4 I (3C " )6 ' !
+ * Z 1 1+ 2.5 + 2.4.5.7 + 2.4.6.5.7.9 + "
= A' (1 - 3c?) e 3ra + 5' (1 + 3oz) e"
■::<.'-
(ii) Change the variable from x to z by the relation 2 5 = #,
and use the same integral as in the last example ; the equation
becomes
d 2 y 4,dy
-t% ? — 25c 2 y = 0,
dz 1 z dz v >
and the primitive is
y = A (1 - 5cz + -^c 2 ? 2 ) e 5M + B (1 + 5cz + ?£c 2 z 2 ) e~ 5C!! .
CHAPTER V. MISCELLANEOUS EXAMPLES 61
(iii) The equation should be
dy 2 dy _ / 2 \
dx 2 xdx \ x 2 ) "'
the primitive of which is, by § 112,
x y =x '"(ldx)~ (Aenx+Be ~ nx >
Of the equation as given; the primitive is
x%y = AJp (nx J- 1) + BJ^ (nx J^l),
where ix=^J—l.
Ex.2, (i) y = x{AJ 2 ^xi) + BY s (^xi)} ;
(ii) The series x* S AmX**™ satisfies the equation if
m =
( f i + m-l)(fi + m-5)(fj- + m + 6)A m + (i J , + m + lfA m ^ = 0,
and 0*-l)(/*-5)0* + 6) = 0.
The value p — — 6 gives a polynomial ending in x~- ; the value
fj, = 1 gives a polynomial ending in x i ; and the value yu. = 5 gives
the series
( 72 72.32 _ 2
L_ 1.5.12 a, + 1.2.5.6.12.13 a;
7 2 .8*.9 3
x 3 + .
1.2.3.5.6.7.12.13.14
(„ -I)(a_«) ( t-I)(t-4)
V + - 7 f * +
(a-2)(g-5)(ft-2)(&-5)
+ ^ 8 , «,■+...
D »fi (a -6 + 1) (6-1)
(a _ 6 + l)(q-6 + 2) ( 6-l)(6-2) _
+ - -2!(6 + c-l)(6+7=2r W - '
nr L (o + c + l)(c+l)
+ C *\ 1 - l( & + c + l) g *
+ c + l)(a+c + 2)(c + l)(c + 2)
+ 2!(6 + c+l)(6+e + 2) w;
62 CHAPTER V. MISCELLANEOUS EXAMPLES
Ex. 3. The series-primitive is
f x 2 x 3 )
+ BxPi { l+px + p(p + 3)^+p(p + 4 ! )(p + o) — +...[.
For the second part, change the variable by the relation
z 2 = 1 — 4# ; the equation becomes
{ l-^ + 2(p-l)zf z -p{p-l)u=Q.
The primitive of this is
u = C(1-z)p + D(I + z)p.
To compare the primitives, let *■ be small ; then, approximately,
z = 1 — 2x ; the coefficients of x° and x p give respectively
2*>D = A, 2pC = B.
2i 2i
p. 201 Ex. 4. Take y = — ^ cos w ; then a; = cos, 3«. Now
so changing the variable to x, we have
rf J (_ 4 sin ' 2 3w) ~ J 2 *^ 3 cos 3w + 2/ = °>
that is, (i^ 2 + ¥ v) ^ + i* ^ - *fey = o.
Ex. 5. The primitive of the final equation is
£#. 6. Take *' = 9*, 3/' = g-rrz/ ; the equation becomes
'&+«-*>%-<-<>■
The primitive of this is
y' = A' (1 - i x) + B' (1 + JaO e"*',
leading to the result.
CHAPTER V. MISCELLANEOUS EXAMPLES
63
d s z
Ex. 7. The primitive of -j— + q 3 2 = is
z = Ae-i x + Be h - qx sm(^3x+C);
d fz s
then y — . .
17 a* Kxi
leading to the result.
Ex. 8. As in the solution of the second part of Ex. 2, § 90,
prove that u = (1 — 2a# + a 2 ) - ™ satisfies the equation
a „-,|| (1 ^ ri gj +(1 .^- l3 l(„» + ^).o.
Then substitute u = ty m d m , where ?/ m is independent of a; the
result follows.
Ex. 9. Write £T = {P n (cos 8)}* = P m 2 (a;). We have
d („ „.dU)
dx
and therefore
d
|(1-^)
dx
= 2(l-x*)P n '"--2n(» + l)U,
dx
da; [ v
d*
+ 2«(n + l)-£;{(l-rf)ff}
c&c
Hence
d
dx
r o-
--'ii* 1 -
-«%\]
= 2£{(1-W1
= -4(l-* 2 )P„'7l(ft + l)P 11
~ , , w- „ dU
= -2n(n+l)(l-x*)j x .
-! |., M » r I) (1-X*)~-XU 0.
Take a; = cos ; the last equation becomes
£a\ 10. By Ex. 5, (iii), § 90, we have
(2m + 1) xP n = (n + 1) P n+1 nP„_j ;
nnd by Ex. 3, (a), § 99 (see the proof, ante, p. 55), we have
(2k + 1) xQn = (« + 1) Qn+i + nQn-i-
Hence (n + 1) (P«+i <3« - Q»+ijP») = M (-PnQn-i ~ Qn^V-i)
= Pi Oso — Wi P) >
64 CHAPTER V. MISCELLANEOUS EXAMPLES
on repetition. Now, by § 99,
Qi = i*log^— j--l, Q,, = -Jlog^— -;
hence P 1 Q -Q 1 P =1,
and so (n + 1) (P„ +1 Q„ - Q B+1 P M ) = 1.
B, 202 Ex. 11. Differentiate the equation
(l-*-) & -2» E + »(«+l)«-0
m times, and write y = d m u/dx n : we have
(1 - * 2 ) ^ - 2 (m + 1) ar ^ + (n + m + 1) (n - m) y = 0,
This is the primitive if m < n ; but, if m>n, the first term
vanishes.
When m > n, we use the result of Ex. 2, (i), § 99. From the
foregoing, a particular solution of
(l- a -)g-2( n + 2)*g-2(» + l)y»0 I
being any constant multiple of , n+1 " , is (* 2 — l) - ™ -1 . By § 65, the
primitive of the last equation is
y = A O 2 - 1)""" 1 + B (x* - 1)^- 1 f X (x 2 - l) 2 dx.
Now differentiate the last equation the remaining m — n — 1 times,
and we obtain the required equation; and therefore, when m >n,
its primitive is
Jm—n—i fJm—n—i ( rx )
A J^=i K* 2 - 1)— 1 } + B ^^ {(*" - 1)— > J (* 2 - 1)- (fej .
##. 12. In the analysis at the beginning of § 107, take m = \k ;
the result follows.
Ex. 13. Let y = (1 — a: 2 ) m ,z ; the equation in z is the equation
given in Ex. 11. Hence the primitive.
The primitive can also be expressed symbolically in the form
y = AJJ. . . P n dx™ +BJJ... Q n dx™,
the first part of which can be evaluated at once.
CHAPTER V. MISCELLANEOUS EXAMPLES 65
Ex. 14. First change the independent variable by the relation
x — 2 2 ; then the dependent variable by the relation yz n ~^ = u ; the
equation for u is
dht. 1 du f, (n - l) 2 )
dz 1 z dz \ z
Hence the result.
■u = 0.
£#. 15. (This result is somewhat important, as the equation
contains no fewer than four disposable constants.)
First change the independent variable by the relation c^x m = rnz;
then the dependent variable by the relation u = yx% '" ~ l > ; and
use the given value for fj,. The equation becomes
d 2 u 1 du /, u, 2 \ .
Hence the result.
Ex. 16. From § 103, (iii), we have
^{x m J m (x)} = x m J m _ l (x),_
so that j t {$ m J m (**)} - i«* (" " x ) J^ (**),
and therefore ^ {**"* J m (**)} = ^ /. («*).
Further, from § 103, (iii), we have
d
, l ar tl J n (*)} = - x~ n J n +i («),
d
so that ^{r*V„(ti)} = -iri( n + 1 )j fH . 1 («4);
thus g. { j, (( i)} = (^p *-*-/.(«*>
Consequently (- 4*)» ~ {<1»» J„ (i*)} = ^ "V™ («*).
Now let t = — 4«ar, where a"* = 1, so that a has m values ; the
equation becomes
F. "
66 CHAPTER V. MISCELLANEOUS EXAMPLES
d m y
Hence the primitive of the equation x m ,— a „ = y is
, m-1 , ,
y = x* m X [A p J m {2(- ap x)i} + B p Y m {2(-a p x)i\],
p=o
the summation being over the m different values of a.
Similarly for the second part.
p. 203 Ex. 1 7. (i) Change the variable by the relation z = e x .
(iil Change the variables, first by the relation xz=\, then
by yz = u ; the result follows.
Ex. 18. The relation
"" n t ("J —n t **
~fa J ~n ~ ~fa J n = — Sm »T
is established in § 106. But (§ 103)
dJ n _ n j j
A T -~ dn + Jn-i,
\A11As iki
and (when n is not an integer)
dJ^g _ _n r j
j — *J —n " i—7i )
OjOC cc
2
and therefore J- n J n - x + Ji- n J n — — sin nir.
irx
For the second part, use the result of the example in § 106
(ante, p. 58), viz.
j dx n ^ y dJ n _ 1^
n dx dx x :
together with -^ = -J n -J n+1 , -^ =- Y n - i n+1 ,
verifying the latter from the expression in § 105. The result is
V T —TV — -
and not as stated in the text.
Ex. 19. Take u = e~$ dx y; the equation for y is
and of its first derivative.
For the example, take m = 2, p = 2 ; the primitive is
, /l d V sin (lex + a)
y = AxH-- r )
\x dx) x
= A{(3 — k 2 x*) sin (he + a) - 3kx cos (kx + a)}.
5—2
68 CHAPTER V. MISCELLANEOUS EXAMPLES
[Note. The immediately succeeding examples belong to the
type of symbolical solution of ordinary linear differential equations
— a process that was much developed by Boole (and by Carmichael
in his Calculus of Operations). Boole's results are contained in
chap, xvii of his Differential Equations, and in chap, xxx of the
Supplementary Volume.
The whole of this method belongs to a very formal stage of the
solution of differential equations. It is less used than it was in
Boole's time ; for it imposes limitations upon the constants that
occur in the equations, and these limitations are often not satisfied.
In the latter event, the integration of the equations is obtained by
means of infinite series that cannot be expressed in "finite terms."]
Ex. 23. (i). In the equation (1 — ax 2 ) -j^ — ax ~- — cy = 0,
change the variable to t by the relation -=- = (1 — ax 2 )^ ; the
equation then is -^f — cy = 0, and is integrable in finite terms
whatever c may be.
Now differentiate the equation (1 - ax*)-^— bx-f- — cy = 0,
CLOD CLflQ
and write y 1 = -^-\ we have
If the former is integrable in finite terms when b/a is an odd
integer, so is the latter, for (b + 2a) /a then also is odd. It has
just been proved to be so integrable when b/a = 1 ; hence the
result.
d
(n) and (iii). When 6 deriotes x -=- , and when a and /3
denote the roots of the quadratic
ap (p -l) + bp + c = 0,
being* jr
l^d-k)'.^
(-!)•
the equation can be expressed in the form
% = a{e-*){6-P)y.
* There is a misprint, as regards the sign of icja, in the text.
CHAPTER V. MISCELLANEOUS EXAMPLES 69
Then, taking x = e f , the equation can be expressed in the sym-
bolical form
6(0-1) = e *a (6 - a) (d - 0) y
= a(6-2-a)(6-2-/3)e 2t y.
By the application of Boole's propositions in the volumes quoted
in the preceding Note, the results are established.
Of course, the use of the symbolical calculus is not the only
way of establishing the results. Thus if
y = l,A m x m
is to satisfy the equation, the form
g = a(0-a)(0-/3)2,
gives 2m (m — 1) A m x m ~ 2 = a% (p — a)(p — /3) A p x p ,
from which it follows that there is a finite integral both when
the initial term is A^ and when it is A x x, as it can be, provided
a or /3 is an even integer. This result ensures the solution of (iii).
Again, by taking an integral beginning with x a or x& and
proceeding in descending powers of m, we can obtain an integral
in finite terms when a - /3 is an even integer — a case not included
in (ii).
Ex. 24. See Pfaff, Disquisitiones Analyticae, and Boole (Diff.
Eqns., cit. sap., chap, xvii, § 7), where the result is established.
Ex. 25. The first expression is obtainable by substituting p. 205
x-p (A + A 2 x* + A i x i + ...)
and determining the ratios of the coefficients A : A 2 : A t : ... so
that the equation is satisfied.
For the second and the third, take
u — ye**,
where /j?= a-; the equation for y is
f iy + 2 „ d y = p(p +1 \ .
dx 2 dx x 2
Substitute x~p (C'„ + C x x + C 2 * 2 +...),
and determine the ratios of the coefficients G„ : C^ : C 2 : . . . as in
the preceding case ; the results follow as stated.
70 CHAPTER V. MISCELLANEOUS EXAMPLES
When p is not an integer, the three series (after substitution
for e m and e~ ax in the second and the third expressions) have the
same coefficients for powers of x~~ p and oc~ p+1 ; hence (§§ 83, 84) the
three series are equal.
When p is not an integer, take q — — (p + 1) ; the differential
equation is unaltered ; and so any one of the three expressions, with
— (p + 1) substituted for p, is a second and independent particular
solution.
Ex. 26. The first two forms, due to Boole {I.e.), follow from
the propositions just quoted in Ex. 24.
The form due to Donkin is only another mode of writing the
result on p. 199 of the text as
/ d l\ m
y = x m I j- -) {A'e nx + B'e-> IX ).
But the formal calculus of operations is very elaborate for the
solution of special linear equations.
Of course, there are other ways of solving these equations.
When y is given by
yx m = A'e ax + B'e~ ax ,
it satisfies the equation
x 2 y" + 2mxy' + m {m — 1) y = aVy.
For equations of the form considered, it is convenient to take
either z = x* (which is the substitution in the text) or z = a; -2 .
With the latter, the equation for y is
4z 3 ^ + (12g - 4m + 6) & ~
+ {I2q (q-l) + q (12 - 8m) + m (m - 1)} 2T - a 2 T = 0.
Next, as regards the given equation
d *y „*.. _ i
dx 2
- a2 y = - 2 p (p + !) 2/>
X
take y = w** ; the equation for u is
x- j— 2 + 2\x j- + X (X - 1) w — aVa = |> (j> + 1) m.
Changing the independent variable from x to z as before by the
relation z = x~ 2 , we have the equation for u in the form
4z 3 j{ + (6 - 4X) z* ~ + {\ (X - 1) - p (p + 1)} zu - ahi = 0.
The equation for u is the same as that for T if
12<7-4m + 6 = 6-4\,
that is, X = m — 3q, and if
12? (q-l) + q (12 - 8m) + m (m - 1) = X (X - 1) - p (p + 1).
There are two values of m in terms of q. For the first
m = 2q — 1 ; then X = — g — 1, and q, a positive integer, = p. For
the second m = 2q — 2; then X = — q — 2, and q = p + 1. When
these results are combined with the transformations, the first
solution of the equation is
y = %-p-i (a? -j-Y {x- 2 p +1 (Ae ax + Be~ ax )) ;
and the second solution is
/ (j \p+i
y = arP-» (ik 3 t- ). {ar-^ (^e a9; + Be-™)}.
Ex. 27. For the first equation, let y = ue~ ax ; the equation for
u is # -3- + {m + re — (a — /3) .z} -5 m (a - /3) u = 0.
72 CHAPTER V. MISCELLANEOUS EXAMPLES
The quantity vf, = x~ n e x{ °-~®, satisfies the equation
xw" + {n — (a. — ft) x\ w' = ;
when this is differentiated m — 1 times, it leads to the foregoing •
(jrn-i w
equation with u = -y _ s"-f > so a particular solution of the original
equation is d m_1
v = e~ ax t \x~ n e x {a -~^\.
J dx m ~ l l '
That original equation is unaltered if to and n, and a and /3, are
simultaneously interchanged ; hence another particular solution is
rln-l
« = e -P*- \x- m e x ^- a) }.
Hence the primitive.
For the second equation, let y — e^ x u; the equation for u is
u" + xv! + (m + l)u = Q.
Taking the equation v/' + xw' + w = 0, the foregoing equation
gives u = -j-~ ; the primitive of the equation in w is
w = Ae-i x2 + Be-i x '-fe h - x2 dx,
for e~^ x is a particular solution. Hence the primitive.
p. 206 Ex. 28. The differential equation of the given family is
1 dP n d/j, _n+l dr
T n ~d^ d~e~~^r r dd'
and therefore the differential equation of the orthogonal trajectory is
_1_ dP n dp__, dd
P n dfi dd~~~ {n+ )r dr
= ~ ( - n+1)r £d f ,-
Thus, as ju. = cos 8, the equation is
By means of the properties in Ex. 5, § 90, this can be expressed
in the form
n dr = {2n + l)P n = 1 ( dP n+1 _ dP^
r dfjL P n+1 - P n -i P n +i — Ph-i V d/j. d/u,
by (iv) in that example ; hence the result.
CHAPTER V. MISCELLANEOUS EXAMPLES 73
Ex. 29. The first part is mere substitution and differentiation.
For the second part, let a and /3 be the roots of
The primitive of the equation is
s
y= 2 K r (x -a) m r{x -$)*>■
r=l
where m,, m 2 , in s are the roots of
in s — 3m 2 + 2m = -j (a — /3) s ,
and ??,. = 2 — m,. ; the quantities K lt K 2 , K 3 are arbitrary constants.
Ex. 30. (The solution of the equation, in the given form,
requires a knowledge of the properties of elliptic integrals.) Let
x? = £ ; the equation becomes
e^ r 2£ 2(l-£)'
After § 110, substitute
the equation for u is
(
(
that is, if f + f ' = 1
„„ 1 efe
d 2 w. 14 1 M _ .
dj* + | df + 4 f(l-?) ~
wf 2+ 4r| + «=o.
The primitive of this equation will hereafter {§ 144, Ex. 3, (i)} be
proved to be
u = AE+B(E'-K'),
or, in the notation used by Glaisher*,
u = AE+BJ f .
Moreover, it is proved (I.e.) that
dE 1 1 dJ'_ 1
^ = -2l' /= -2l (j& - if) ' dF- - 2A^ '
hence the result.
* Qwart. Journ. Math., vol. xx, p. 318.
CHAPTEE VI.
p. 208 § 113. Ex. 1. We have
d»F(a, & 7, x)
_ a(a + l)...(g+ W -l)/3(/3+l)...(/3+ w -l) Et/
7 — TT^ ~i — ! T\ ' («+ w > P + K, V+n, x).
7(7+1). ..(7+w-l) "^ '' 7
The series diverges for # = 1 if 7 < a + /3 ; so, in that case,
7 + w<(a+w) + (/3 + n),
and therefore every derivative diverges for x=l. Even if the
condition for the original series is not satisfied, from and after
some value of n the condition 7 + n < (a + n) + (/3 + ri) will be
satisfied; and then all the corresponding derivatives diverge for
x = l.
Ex.2, (i) sint = tF(a, 0, §, -^J-) , «_> oo , /3->oo ;
(ii) sinni= ?! sin t . F (| — ■£»?, £ + £n, §, sin 2 1);
(iii) cos??i= (1 + tan 2 t)~^ n F (- %n, \ - |ra, f , - tan 2 £).
p. 224 § 125. #x. 1. We have
, . 11.,, 1.3 1.,
t = smt + ^.- 6 sm s t+ -^—r . - sm 6 1+ ...,
IS 2.45
,, 1 ,111.81
sothat i7r = l+_. _ + _._ + ...
= n(i)n(-i)
= 2 {n (i)} 2 .
Hence II (£) = -^ *Jtt.
Ex. 2. We have
11 ( - Z) " ^ (l-^-lV-. *(*-*) **'
TT/ llT - 1 • 2 . 3 . . . k
CHAPTER VI, §§ 125, 127 75
and therefore
n(-*)II(*-l)=Lim '
*(i-£)d- Jm :.
37'" I (k-iy\
= 7r cosec ZTT.
Ex.3, (i) F 1 (a,0,v)F 1 (-a,0, y -a)
_ II (7 -1)11 (7-8-/8-!) n (7 -a-l)n (7-/8-I)
n (7 - o - 1) n (7 - /8 - 1) - n (7 - 1) n (7 - a - /8 - 1)
= 1;
(ii) Interchange a and /3 in the foregoing example : or use
again the formula at the end of § 125.
Ex. 4. Take the case n = 2 : we then should have to prove
2 2z + * IT (z) IT (* - $) = (2tt) n (2s).
Now TI (>) = Lim , -VT" — '^,'" , r\ & z >
7 t^.^+1) (5 + 2). ..(«+&)
tt/ n t- 2.4.6...2& 7 . ,
n (*-*)- Jam (2 , + 1)(& + a)(2s ^ ab _ 1) * *
^■S (2, + l)(2, 2 + 3 2i: 2 (^ + 2 T) <**
U(2z) . (*+l)(*+ 2). ..(* + *) ...
so „, ,„. — -r- is equal to t^ — j— ~ -rr ^ ^ , multi-
II (z)U(z- £) ^ (2s+2)(2.z + 4)...(2,? + 2&)'
plied by 1.3.5 ... (2k- 1) .
r;"2T37^ l k '
,(l--i)(2-i)...(A--i)
that is, it = 2 M &4 1.2.3...&
■ = — 2 M .
n(-i) Vtt'
Hence the foregoing special result.
The general result can be obtained in a similar way. See
Gauss, Oes. Werke, t. iii, pp. 149, 150.
§ 127. Ex. 1. The relation in § 126 between F 1: F 2 , F 3 will be p. 227
denoted by (a). The relations in § 127 between Y lt F 2 , F 4 ;
7 1 , r„ F 5 ; F, F 2 , F 6 ; will be denoted by (/3), (7), (8) re-
spectively.
76 CHAPTER VI. § 127
(i) Eliminate Y 2 between (a) and (/3); taken as a relation
between y lt y 5 , y 7 , it is valid in a range < x < 1 ;
(ii) Eliminate Y 2 between (a) and (7); taken as a relation
between y lt y 5 , y it it is valid in a range < x < 1 ;
(iii) Eliminate F 2 between (a) and (8); it is not valid for
purely real values of x (see the note in the text, p. 220);
(iv) Eliminate F 2 between (/3) and (7); taken as a relation
between y x , y 7 , y 9 , it is valid in a range < x < 1 ;
(v) Eliminate F 2 between (/3) and (8); it is not valid for
purely real values of a; (again see the note quoted above);
(vi) Eliminate F 2 between (7) and (8); taken as a relation
between y 17 , y 10 , y s , it is valid in a range — 00 < x < — 1.
p. 228 Ex. 2. Eliminate Y x between the relations denoted by (a) and
(/3) in the preceding example.
Ex. 3. There are twenty relations in all, between the groups
given by selecting threes out of Y lt ... , F„. Four are given in the
text; seven are given in the preceding two examples; so nine
remain. They are as follows : —
(i) Y, = -A(-iy-yY 2 + BY 5 ,
where
U(-S)U( a + /3-y) II(-/3)n( g + /3-7)
II (a- l)II(l- y ) ' ° n(a-/3)II(/3-7) '
(ii) Y 3 = -A'(-iy-yY 2 + B'Y s ,
where
n(-a)n(a + /3- 7 ) n(-«)n(a+/9- 7 )
II 08-1)11 (I-7) ' -" n(a- 7 )n(/8-a) '
(iii) Y i = -A"(-iyY 2 + B"Y 5 ,
where
n(tt- 7 )ii(7-tt-fl) „„_ n( a -7)n(7-«-/8)
11(1-7)^7-^-1)'^' n(-«)n(«-/3) '
(iv) Y i = -A'"(-iyY 2 + B'"Y c „
where
r ,_n(^- 7 )n(7-g-/8) n(/3-7)n(7-«-/3)
n(i- 7 )n(7-a-i)' n(-/8)n(/8-o) '
(v) -(-l)vF 2 =CF 5 + i)F ,
where
n(i-7)n(/3-a-i) n _ n(i-7)n( g -/8-i)
n(- a )ii(/3-7) ■ " n(- / a)n(«- 7 ) >
CHAPTER VI. §§ 127, 128 77
(vi) F 5 = i(-1)»F 3 -M(-1)^7 4)
where II( 7 - «-/?) n( a+/ 8- Y -l) !!(« -£)
n(-/3)n( 7 -/3-i)n(« + /3- 7 ) •
n( a + /8-7-l)n(a- / 8) .
II (a - 7) 11 (a -1)
(vii) F 8 = Z'(-iy%-M'(-l)7-«F 4 ,
n( 7 -a-^)n( g + ^- 7 -i)n(/3- a )
n(-o)Il(7-o-l)n(a + /tf-7) '
n(a4-/3- 7 -l)(/3-«) .
n(/8- 7 )n(/S-i) '
(viii) F 3 = P (- 1)- F, + Q (- I)"* 5 F„
where
n(a + ff- 7 )n ( ff-« - 1) _ n(« + /3- 7 )n(«-/3-i) .
n( j 8- 7 )n( i s-i) ~' w n(«- 7 )n(a-i)
(ix) Y 4 =S(- l)-r* F 5 + T (- 1)->+» F 6)
where
n(7-a-/3)n(/3-«-l) n(7-a-/3)n(«-^-l)
11 (-a) II (7- a- 1) ' n(-/3) 11 (7-/3-I) "
Reference should be made to Goursat's memoir, quoted in
§ 134.
128. Ex. For the first part, we have (p. 229 of the text, p. 230
8) an expression for
stitute from the relation
§ 128) an expression for y 5 -p - £ = i.
leading to the equation stated, which therefore can be solved in
finite form.
p. 239 Ex. 2. In case III, we have
^ 2 = i. ^ = h * 2 = f;
these equations are satisfied by
a =h £ = *. 7 = 1.
leading to the equation stated, which therefore can be solved in
finite form.
Miscellaneous Examples at end of Chapter VI.
p. 240 Ex. 1. It may be assumed that a and b are unequal; so, to
secure convergence, we take a > b.
Now (a 2 + 6 2 - 2ab cos )~ n = ar™ 1 - - e*»J h -_)~ n cos r^tf^.
For (ii), we take
^ = (^T6T ' o I ^+& cos * J cos r< ^'
and expand in powers of 2ab/(a 2 + b 2 ); for (iii), we take
1 f 7 f 4a6 "l ~ n
^'•=(^T6rJo t 1- (^67 2COs2 M Cosr *^'
and expand in powers of 4hkf = e 1 - ihk .
(ii) For the second result, the corresponding linear equation is
^-fc^ + ku
(iii) For the third result, the corresponding linear equation is
3m .,„du „ , _
5T + 4& 2 ^ + 6^ = 0.
oh ok
The forms, required for (ii) and (iii), are deduced as for (i) above.
Ex. 7. (i) The subsidiary equations are
doc, dx 2 ,
= dz.
Xj — ^Jj X 2 - x 2 X s
Let 2/i = *]y3, 2/2 = #2?/3: where y 3 is determined by the equation
dy„ „
-£ = y 3 X 3 = a 3l y l + a 32 y 2 + a 33 y 3 ;
then -^ = y 3 X, = «ii2/i + a 12 y. z + a 13 y 3 ,
du 2 xr
-?- = y 3 X 2 = a 2X y x + a 22 y 2 + a w y 3 .
150 CHAPTER IX. MISCELLANEOUS EXAMPLES
These three equations have been solved in Ex. 3, § 176 (p. 353 of
the text): two independent integrals can be taken in the form
h^i + m^ + n, = A
l 3 x x + m 3 x 2 + n 3
l 2 x x + m 2 x 2 + m 2
, = B (log' z) k '~\
L 3 x 1 + m 3 x 2 + n 3
The primitive is
Z^ + m^+n . = F {l 2 x 1 + m 2 x 2 + n 2
4*! + m s x 2 + n 3 8 (4*i + «? 3 «2 + %
the constants being given by equations that, only in notation,
differ from those given in the solution of the example quoted.
(ii) Let
Sl = X 2 p 3 X 3 p 2) ^2 ~ ®&Pl <^1 Ps> ^3 = ffilPz X 2 Pi.
The subsidiary equations are
d*! ckc 2 cfe 3
_ — dp! _ — dp 2 _ — dp 3 _ dz
— a 2 p 3 s 2 + a s p 2 s 3 -« 3 piS 3 + «ii>sSi - chp^i + a 2 Pi*2 1
Then — 2 = (a 2 - a 3 ) s 2 s 3 = fts^,
-5— = (a 3 — aj) 536'! = P2S3S1,
j^ = ( a i- a a) s i s 2 = /3 S Si*2 ;
so if t- = 2s 1 s 2 s 3 ,
we have 2s, -=- = /3, -=- ,
dz dz
and so s x 2 = ftw 4- A lt s 2 z = /3 2 u + A 2 , s 3 2 = /3 3 u + A 3 ,
where a l A l + a 2 A 2 + a 3 A 3 =\.
Now x x Sj + x 2 s 2 + # 3 s 3 = 0,
so that u is a function of x given by the equation
x x (ftw + A,f + x 2 (/3 2 u + A 2 )i + x s (/3 s u + A ,)* = ;
and then 2^ + 5 =/{(&« + A) (/3 2 w + A 2 ) (/3 s u + A 3 )\ ~§ du.
There are three arbitrary constants.
CHAPTER IX. MISCELLANEOUS EXAMPLES 151
Ex. 8. (i) One integral of the Charpit equations is
Hence (p-qf= \{x — yf- A ;
and therefore
z-B = %m(x + yy + A}i(dx + dy) + lf{±(a>-yy-A}i(da ; -dyy,
(ii) z -xy =lax 2 + l^- + b;
(iii) *=!(a + l)^ + i(3 + l)2/ 2 + &;
(iv) z — x-^x.^ = Ax? + Bx 2 3 + Cx s * + A',
where ABC = J T .
Ex. 9. The equation of the required surface is
z 2 = a (* 2 + xf).
Ex. 10. Denote the two values of dy/dx at any point by t lt t 2 . p. 468
Then it — i
so that the two curves are orthogonal.
The product of the curvatures
/ ' t'
(1 + tf) 1 ' (1+V) 1 "
Now « 1 = «+(l+^p. * 2 = 2-(l + .z 2 )i;
so t' = {1 + *(1 + «•;"*} (l» + S(x 2 + y 2 ).
Ex. 13. Surfaces orthogonal to a given family of curves satisfy
the equation
dx _dy _ dz
p ~ q ~^1'
where dx, dy, dz belong to the curves at any point, and p, q, — 1
belong to the normal to the surface.
We find p — coth x tanh z, q = coth y tanh z ;
so the surfaces are sinh x sinh y = A sinh z.
Ex. 14. We have
g^ = xy'fz- c t>™i r y + yfxz ~ zfxy,
and so for ^— , ■=- . The sum of the three vanishes.
dy dz
For the converse, take two functions and t|t determined by
the equations
4> y tyz - 4>ztyy = w, 4>zi r x-4>^z = , v;
these functions satisfy no other condition ; then
dw _ du dv
dz dx dy
= 4>xz^y ~ yzfx + xir vz - y -f xz
= fa,(xfy-yi r x)>
whence the result.
Ex. 15. (i) If the two equations have a common solution
other than u = constant, then
CHAPTER IX. MISCELLANEOUS EXAMPLES 153
Also du must be an exact differential; hence
( YZ' - ZY') d.v + (ZX ' - XZ') dy + {X Y' -YX')dz =
must be reducible to an exact equation du = 0.
Let its integral be (x,y,z)}.
(ii) The two given equations have no common solution p. 469
other than u = constant.
Ex. 16. The equations, subsidiary to the given equation
F=0, are
dp 1= __dp. 2 dp 3 = ^P4 = dps
3p„ + 4p~ 2p, + 5p s p 5 ~ pi
Pi
_ d.i\ dx. 2 dii: A dxt
— 1 - Sx 2 - 2x 3 - a- s p 5 - 4a-, - 5x 3 + x s p B p£
dXr,
- a\ - .r 5 ( p. 2 - p 3 ) - 2 & x s
P*
Integrals of these equations*, consistent with the original equation
and with one another, are
(p-2-p 3 )e*'=Ai
(p 2 +2p 3 )e' ! *>=A,
£? e p s -P2 = A t
P4
The corresponding complete integral is
z - A = ^A 1 (2x 2 -x 3 )e' x ' + A 2 (x 2 + x 3 )e- yx ' - A 3 A t le^^'dxi
+ AAog (x 4 + A 4 ;r 5 e A i e ~*').
For the second part, make the transformations
Z = p l x l + ... -f p^r a -z,
x r = P r , p,. = X r , (for r = 1, . . . , 5).
* They can be chosen in a variety of ways, and lead to different forms which
functionally are included in one another.
154 CHAPTER IX. MISCELLANEOUS EXAMPLES
The transformed equation is
X, + (3X 2 + 4X 3 ) P 2 + (2X 2 + 5X 3 ) P 3
x;
+ x 6 p 4 + (x 2 - x 3 + J?) x 5 P 5 = o,
a linear equation. The subsidiary equations are
dZ 1 dX, dX s dX t dX 5 dZ
M„ =
3X 2 + 4X 3 2X 2 + 5X 3 X 5 W Zii _ Zi + */\ ~ z a'
and five necessary integrals can be taken in the form
u-l = Xi = a lt
X 2 4- 2X 3
(X 2 -x 3 y~ a *'
•"■5 T- _ V
3 = X e = tta '
M 4 = - log X 4 - je*i-*» d log (X 2 - X 3 ) = a t ,
The most general integral is
<& = m 5 — F (u lt u 2 , u 3 , w 4 ) = 0.
In order to return to the integral of the original equation, we
use the relations
Z = x 1 X 1 + ...+ # 5 X„ - z,
Y + dT = ' (for r = l, ..., 5).
Ex. 17. (i) z = A 1 x l 2 x 2 - A x x^ - s— (« + b log A) + A s oo£ ;
(ii) {2zf-- A 1 = A 3 x 3 -^A 2 x^ + A 2 x 2
-14^ + 2 A [\A{ + 0^0- AM*.
Ex.\%. z-B = A (x 2 x? - x, x 3 2 ) + log (x 2 x t ).
Ex. 19. (i) z-C=A log {(«, + x 2 y + (x s + x t f)
+ B log \{x, - x 2 f + (x 3 - x 4 y] ;
(ii) z - C = A log {(x^ + x}) (x? + x?)}
+ B tan- (°^1±^\
\XiXi - x 2 xj
CHAPTER X.
§237. Ex. (i) * = + (!,)+ je-* M U (x)+j Ne^ [ dy\dx; p. 473
(ii) z = (f> (x) + jei mx U (y) + NJJ Mdx dx\ dy.
§ 240. Ex. This is mainly a complicated exercise in deter- p. 475
minants. Write
_ d*z d 2 z
dx, 2 ' dx 2 dx 3 '
and so for the others : also
and so for the others. Taking derivatives of F (, yjr, ^) = and
,. . ,. dF dF dF
eliminating ^- r . ^—r . 7— . we have
° d(p ' ay Bx '
dcf> d±
& + Zu 5^~ + Zl2 ^T + Zu ^ ■
3$ 3<£ 3$
9p! 3p„ 3^ 3
< < 33 + Zu dJ 1 + Z23 dp 2 + Zss dp 3 <
= o.
As every first minor in J ( "' ) vanishes — and therefore also
\Pi,Pi,pJ
J itself — the terms of the second order and of the third order
in z,„ z u , ..., z 3S disappear. The coefficient of z n is R„ where
R,=
d(f) difr dx
dp, ' dp, ' dp,
$1, tyl, %2
3, ^3, %3
156
CHAPTER X.
I 240
and the coefficient of z w is R&, where
XI23 =
d d-yfr dx
dp 2 ' dp. 2 ' dp*
+
3
dp,'
3i|r
dp 3 '
3_X
dp s
4>i, fi, Xi
4>3>
•^•3.
X:
4>i> ^2, X"
2, fa, X*
08. fa, %3
In V, let the minor of fa be denoted by ,, and so for the
other constituents. Then
dp 1 dp, dp x
3p 3 3p s dp, dp 2 *dp 2 dp 2
and so for the other quantities.
Consider the determinant
O = 2_Z?i , -Kl2) Ry,
■^12) ^-t^2 3 -^2
J? ]3 , -R ffl , 2R 3
dpi dp* dp s
then because of the properties of J 1 ^'*' X \ we h ave
opt tyi 9p 3
xjk+A + ^-o.
dp* 9p a 3p 3
Multiply the rows in ® by X, p, v, and add the first to the second
and third ; we have
Take
/Xl>© =
opi 9/>i c^
cp 1
■dp,
dpi'
q d± + yj± x 3*
dpi 9pi 3p 2
chapter x. §§ 240, 250 157
The right-hand side is
<*>„ %, x,
and therefore vanishes. Hence ® = 0, and therefore
iij.R.23 2 + J? 2 i? 31 2 + R 3 R ]2 2 — 4>R 1 R 2 R 3 — R l2 R w R zl = ;
and the equation is
-"l-^ii i • • • ~r -it23^23 T" • • • = ' •
§250. -£«. 3. (i) When k is not unity, there are two intermediate p. 490
integrals
1 - -j^ P =/i \y + a (1 - »}>
? - -^- p =/. {y + * (i + «} ,
where £ 2 = 1 — k. These coexist (§§ 245-247) : so we deduce the
values of p and q, and find
z = F{y + a(l-l)x} + G{y + a(l + l)x).
When k is unity, we have
z = F(y + ax) + xG (y + ax) ;
(ii) The intermediate integral is
*p + yq - ' = (f ) ;
and the primitive is
(iii) The intermediate integral is
(P\.
z = d>
T \q
the primitive is F (z) = y + xG (z) ;
(iv) z = F(xy) + xG(^y,
(v) An intermediate integral is
p + aq = e- 2abx f(y + ax) ;
and the primitive is
z = F (y - ax) + e' !abx G(y + ax).
158 CHAPTER X. § 250
p. 491 Ex. 5. (i) Two intermediate integrals are
p + ay = - 2aF' (q - ax), p - ay = 2aG' (q + ax) ;
and the primitive is
z — qy = F (q— ax) + G (q + ax)] _
y = -F'(q- ax)-G' (q + ax)\ '
(ii) (The equation should have - q + y (s 2 - rt) for its right-
hand side.)
An intermediate integral is
and the primitive is
z = ux-\& + F^ + G(u)
= u + -F'(-) + G'(u)
y \y)
(iii) An intermediate integral is
F{fq-\x\ pq-\y>) = 0.
The integral of this equation is not attainable in finite terms;
but possible separate and non-coexistent integrals of Charpit's
subsidiary equations are given (§ 249) by
p 2 q -\x?=a, pq- ly* = b.
Of the first, an integral is given by
z-A'=A[x(x 2 + laf + log [x + {x> + 2a)*}] + -|- 2 .
Of the second, an integral is given by
z-B' = ^+B{by + ^f).
p. 492 Ex. 7. (i) The only intermediate integral is
V* yj
The primitive is obtained by eliminating a between the equations
a + z = x'F(a) + y*G(a)\,
l=x*F'(a) + i/G'(a)y
(ii) The only intermediate integral is
4>{p 2 q-2,y, q 2 p -3x) = 0.
CHAPTER X. § 250 159
The primitive is obtained by eliminating a between the equations
(a + zf={3x + F(a)}{3y + G(a)} \
3 (a + zf = {3* + F(a)} G' (a) + {Sy+G (a)} F' {a)\ '
(iii) An intermediate integral is
F {.»■ + p(l +p* + q*)-i, y + q(l +p s + q ?yi} = o.
Of the Charpit equations, subsidiary to this equation,
x+p(l +£> 2 + 2 2 )"* = a
is a particular solution ; so, with it, we take
y + q(\+p i + q*)~- = b.
Then we have a solution, from the integration of
dz=pdx + qdy,
in the form (z - c) 2 + {x — a) 2 + (y — bf = 1 ;
and a more general integral in the form
{ x -F( c yr- + {y-G(c)Y+(z- c y = i )
{x-F (c)} F> (c) + [y-G (c)} G' (c) + (z - c) = J '
Ex. 8. Let the equation (/> (x, y, z, a, b, c) = 0, be resolved, so
as to give z=f(x,y,a,b,c);
and note that, owing to the equations x(a, b,c) = and i|r (a, b, c) = 0,
we have b and c as functions of a.
For the envelope of the surfaces, we have to join
df + Vdb + tfdc =0
da db da dc da
with the equation 2=0; and, also,
3/
P=dx'
S =
dy'
For the envelope, the various equations give a, b, c as functions of
x and y ; so, still for the envelope,
dx' 2 x dx 2 dx 3 dx
3 2 / _ da p db p dc
dxdy x dy 2 dy 3 dy
d 2 f n da n db _ dc
~dx~dy + ^dx + ^Yy + ^Fy
t = 4 +Ql dy + Q ^ + Q °dy
(-PiQ, - P.O.) ^ (^) + (P.Q, - P 3 Q 2 ) J (J-
160 chapter x. §§ 250, 252
v, P _ s !/ P _ 92 / p _ 9 !/
wnere ^- S ^ tt > l '~dxJb' Is ~dxdc'
Q=VL Q-»L Q-2L
Hence (r -g) (* -g[) - (.- J£) (. - J£) ,
when in the factors of the second product we substitute the two
values of s, becomes
+(-P.«,--P,<2.)J"(~)-o.
\*j y /
But, because b and c are functions of a, all being functions of x
and y, we have
W, yl \x, yl \x, yl
Hence we have
Accordingly, when the equation of the envelope is taken in the
form
Rr + Ss + Tt + U(rt - s 2 ) = V,
the coefficients satisfy the equation
S^^iRT+UV).
p. 498 § 252. Ex. 2. The equations that arise in the process are
u x u y _ Tj'^x _ rp^y _ TT _ r\
VpUq Up llg
^ + Uy_ R >±g_ T Up + 2s= ^
Resolving these simultaneous equations for
Up Uq
we find that ^-T=X^, ^ - i? = M ^
U p Up u q ^ Uq '
where A. and /x. are the roots of
6*+20S + RT-V=O.
CHAPTER X. § 252 161
When the roots of this quadratic are unequal, there are two
systems of equations, as X and /* can be taken in either of the
two combinations.
When the quadratic has equal roots, the two systems are the
same.
Ex. 3. Write
a / \ %u du . n du du _
. , , , du du cu _ du _
(i) One condition for three integrals common to the system
is p =
then 0- o -)P=A'(T)-A(o-), (p - (c),
O TT
and therefore /3=yfr (c), together with — = 0, i.e.
oc
[x - <$> (c)} ,
A laxity
u v u q
(l + p 2 +q*)
flUy = 0,
r 1 + ^ + ( 1+ (/j,), z-ix(l + ^y=-^(/x);
and the Jacobi condition for coexistence is necessarily satisfied.
When fj, is eliminated, we have the most general primitive of the
equation, <£ and \jr being arbitrary functions.
11-2
164 chapter x. §| 252, 253
Let 8 denote /a + (1 + /*")*, so that - - = ^ - (1 + /a 2 )* ; clearly
8 is a complex quantity. Then we have
y + iz + x8 = <]> (ji) + i^r O) = F (8),
y-iz- x-g = (f> (fi) - iyjr (ji) = G(8);
and the surface is given by eliminating 8 between these equations.
Usually it is imaginary.
The simplest (and the only real) case arises when
F(8) = b+ic + (R + a)8,
G(8)=b-ic + (R-a)- e ;
the elimination of 8 gives
(x - af + (y - bf + (z- cf = R\
(The result is due to Monge, Application de Vanalyse a la
geome'trie, pp. 196-211. See also a note by Forsyth, Messenger of
Math., vol. xxvii, 1898, p. 129.)
Ex. 11. The intermediate integrals of the respective equations
have already been given (supra, pp. 157-159). They can be
obtained by the process indicated in the text.
p. 504 § 253. Ex. 1. The primitive of « 2 r + 2xvs + yH = is (§ 250,
Es ' 3 ' n) -"(*H©-
By the dual transformation (a contact transformation), the equation
becomes q * r _ 2 pqs + p H = 0.
Thus the integral of the latter is given by the relations in the text ;
and it is x + yf (z) = g (z).
Ex. 2. (i) z = xy {F (x) + (y)} ;
(ii) This equation is the dual reciprocal of equation (i), and
two intermediate first integrals are
px-z=pf(q), qy-z = qg(p);
the primitive is obtainable as the eliminant of p and q between
these equations and
pji±vLiA=_\m d P -\m dq>
pq J p 2 J q 2 *'
CHAPTER X. §§ 253, 256
or (what is the same thing) between
165
px + qy — z
pq
= F(p)+G(q)
px — z + pq 2 G' (q) — j '
qy - z + p 2 qF' (p)=0,
(iii) y = zF (x) + xG (z) ;
(iv) The dual transformation leads to the equation
X 2 T - 2X YS + Y*R = XP + YQ,
of which the primitive is
Z = F (X 2 + F*) + G (X 2 + Y 2 ) tan" 1 \ .
The primitive of the original equation is given by
M r+1 = M r ,
N. r+1 = N r - v -P
For the particular equation if 3 = ; so two transformations are
necessary. The primitive is
z = x*G(y) + (2^-^G'(y) + ±G"(y)
+ a?[e- i^ 2 F (x) dx + (\- 2x) [e " **»* xF (x) dx
e ~ % xy2 (x ) xF(x) dx.
p. 510 § 258. Ex. 2. (i) The primitive is
z = Ax + (B - $A 2 ± aA) y + G;
(ii) The primitive is
z = y (-l- A*f + F(x + Ay).
P. 514 § 261. Ex. 2.
y = I \F (x + at) + F (x - at)} + ~- {f(x + at)-f(x- at)\.
P. 515 Ex. 3. (i) z = F(x + iy) + G(x- iy) - °° S ™f C ° S ny - ;
(ii) z = F(2x-y) + G(x-y) + ^ + ^f;
(iii) z = F (ax + y) + xG (ax + y) + \x*f(ax + y) ;
(iv) Let 1, a, a 2 denote the cube roots of unity ; the primi-
tive is
z = F(x + y) + G(x + a y) + H(x + ahj) + ^f^+ rT ^- T - 9 -,
chapter x. §§ 261, 263, 265 167
(v) z = F (ax + y) + yO (ax + y)
+ f 1 (x) dxdx + 1 ]l+ (y) dydy + ^^ f f X (*) dtdt,
where, after the double integration in the last term, x + by is
substituted for t ;
(vi) z = F(x + y) + xO (x + y) + £» 3 + |# 2 <£ (x + y).
Ex. 5. (i) The primitive is p. 516
u = F (x + y, z) + G (x - y, z - y) + \x s yz - -fcxty + T foa? ;
(ii) u = F (x - z, y) + G (2x - y, z) + H(x, Sy + z).
§ 263. Ex. When the method in the text is used, the part of p. 518
the Complementary Function required can be expressed in the form
(D'-aD-PY* 1 ' °
= e 0y .
(D'-aDy+i
= eM {F O + ay) + yF,(x + ay) + ...+ y r F r (x + ay)}.
§265. Ex.2, (i) z = F(xy) + xG(-j + xy\ogx; p. 521
(ii) z = F(xy)+G(l).
y\ x- + y 2 1 X s
x) ~ n-2 ~ 2 nT^3 :
Ex.3, (i) u = xF(£j + x n G:
(ii) u=F fy) +xG (y) + (^±y^.,
v ' \xl \xj n(n — l)
(iii) u = F( y -, - ) cos (n log x) + G[ y -, -J sin (n log x).
x x.
Ex. 4. Change the variables so that
1 + x , y — x
1-x (1 _ ^yk
the equation becomes
dx^ + nU = °'
so that the primitive is
u = F (y) cos «#' + G(y') sin »#'.
168 chapter x. §§ 265, 269
Ex.5, (i) z = F(y-ax) + e- mix G(y+ax);
(ii) z = F (mx + ny) + e~ nx G (nx + my)
x inn cos (mx 4- ny) + (rn? — ra 2 ) sin (mx + ny)
n mW + (m 2 — n'f
inn sin (kx + ly) — (mk — nl) cos (lex + ly)
(nk — ml) {m 2 n 2 + (mk — nlf)
(iii) Adopting the process of § 262, we have the primitive
in the form
x -ftS + A* • -7i/3-Ai
ze 9 7.=ZAe— * + ft, + 2Jfc— ^~ + ^,
where A and B are arbitrary functions of /3, and
A = (A 2 - ah) /S 2 + 2 (gh -af)/3+ g 2 - ac.
Ex. 6. Let /(■&) = A (-sr - Cj) (« — c 2 ) . . . (ot — c y ),
j) being the degree of /(-nr) in ot : then the primitive is
+^^?,...,^)+-
where i^, i^, ..., F p are arbitrary functions of the m — 1 arguments
which are the same for all.
The cases, (i) when there are multiple roots of /= 0, and (ii),
when f(n) = 0, are most simply discussed by taking
x r = e y '\
for r=\, ...,m; the equation then becomes one with merely
constant coefficients. Thus, if c 2 = Ci, the part of the comple-
mentary function is
^{^(S'-'f) + ^G--'l1 log +'
and so for other instances.
p. 526 § 269. Ex. The general solution is
u = (x + at) + ■f (x - at).
Under the conditions, we have
f(x) = (x) + V(x),
-F(x) = '(x)-W(x).
Qj
chapter x. §§ 269, 270 169
The latter gives
- (V(\) d\ = <& (x) - V (x) + A,
where k and A are constants. Thus
(x) = \f{x) + 1 j'F(\) dX - \A,
1 k
consequently
« = $ ( x + at) + NP (a; - a*)
1 rx+at
= *{/(* + at) +f(x - at)} + Ya j^ F(X) dX.
§ 270. Ex. 1. We can take « as the sum of two functions v p. 527
and iv, such that
v =f(x) when £ = 0, and = when x = 0,
f = o = -<*>
1 r 00 f ( a ~ x ) 2 _ ( a + x ) 2 )
= J- e 4a2f _ e 4aH [f(X)dX,
on substituting for/(— X).
We proceed otherwise for the second part*. A solution of the
equation is given by
e - a 2 \ 2 (t - a) C os XX,
and therefore \ e - a '^' ( f _ a ) cos Aa^X,
Jo
* Biemann-Weber, Partielle Different! ' algleichung en, vol. ii, section vi, §§ 40, 41.
170 CHAPTER X. § 270
also satisfies the equation. When t > a, this is
ir' 1 ,. ,_i -
(t-a)*e 4a 2 (t- a)
Further, another solution is given by the derivative of this solution
with respect to *•, which is
nr"-x
a 2 J *
2ati
40, (*-«)"** -4a " (t - a >;
and therefore, as a is not greater than t, another solution is
~ j (t - a)~h ~ 4« 2 («-a) F(a) da.
To find i^, let a new variable /3 be taken in place of a, as given by
0= x - •
2a (t - a)* '
the solution becomes
When x is zero, this is ^— 2 F(t) ; so
and therefore we can take
x [* _a ^
u> = x (* _a ) ' le 4a2 (*- a )$(a)(2a.
2a7r 2 J o
Thus the whole expression* for u is
J ,.00 , (tt-?Q 2 _ (.T + X)8 .
2^#Jof ^--^}/(^
2a7r5-/o (<-X)^
ifo. 2. A solution of the equation is given by
jjf(\ H-) dXd/j,J j cos {(a 2 + /3 2 ) bt]
cos {a (*• - \)J cos {/3 (2/ - /*)} dad/3.
* There is an error in the text, p. 527.
CHAPTER X. § 270 171
Now I cos (a 2 bt) cos {a (.« - X)} da
,00
I sin (a'bt) cos (a (*• - X)} c?a
.' — oo
/TrNij (* -A) 2 . 0-X) 2
= sr, i c os — 7T— - — sin - — r^
\2btJ { 4>bt \bt
and similarly for integrations with respect to /3. Take
\=x+2u(bt)i, fi = y + 2v{bt)^;
the integral becomes
\\f{x + 2w (btfi, y + 2v (bffi} (it 2 + v 2 ) dudv.
For the other term, we proceed similarly from the solution
\\f(\, /t) dXdfif j sin {(a 2 + /3 2 ) bt)
cos {« (x — A)| cos {/3 (y — /*)} dadft.
Ex. 3. (The integral is due to Poisson.) We proceed as in
Ex. 1 by taking u = v + w, where
n/ \ dv . . dw ,. .
v = F(x,y,z), ^ = 0, w = 0, ^=/(/,y,4
when £ = 0.
Special integrals of the equation are given b}'
cos («a<) cos {a (« — A.) + /3 (y — fi) + y (z — v)},
sin (/cat) cos {a (x — \) + 0(y — /j.) + y(z — v)},
where k 2 = a 2 + /3 2 + y 2 . Hence more general integrals are given by
00
V = 1 1 1 1 1 1$> (\, H, v) cos (/cat) cos (a (x — \) + /3 (y — /j.)
-* + y(z — v)} dad/3dyd\dfidv,
00
w = //////* (*• M> ") ^^ cos {a (« - X) + /3 (y - /i)
~ c0 4- 7 (z — v)} dadfidyd\dfj,dr.
It will be noticed that, in form, v is the ^-derivative of w ; so that
it is sufficient to consider w.
172 CHAPTER X. § 270
Now take a = k sin 8 cos , /3 = k sin 8 sin <£, 7 = k cos 8,
k(x — \)= p'sin 0'cos ', k (y— /x) = p'sin 0'sin ', k (2 — i») = p'cos & ,
and let cos B = cos cos 8' + sin (9 sin 8' cos (<£ — 0') ;
then w = - I k sin («erf) eta 1 1 IdXd/xdv'W (X, yti, 1/) /,
/■2ir (-7T
where i" = d cos (p' cos 3) sin 8 dQ.
.'0 '0
By changing the spherical axes, so that the new polar axis coincides
with the direction given by 8', we have h = ; and so
T . sin p
I = 4?r — f- .
P
Take
A. — x = r sin cos cf>, fx — y = r sin sin $, i> — a = r cos 0, p' = rx;
4, y + rsin.8sm P&(x + at sin 8 cos , y + at sin 8 sin d>,
J J
z + ai( cos 0) sin #c£0.
Thus w — when £ = 0. The value of -r- when t = is
/•27T (-7T
w = 2tt 2 d^> ¥ («, y, s) sin 0d0
Jo Jo
= 87rȴ(a!,y,*),
and it should be /(.r, y, z) = 0. Hence
w = j— I d<£ £/"(# + a£ sin # cos 0, y + a£ sin sin 0,
z + at cos 0) sin #d#.
The value of v is obtained as -^- , it f(x, y, z) is changed into
F (x, y, z) ; thus
1 d [^ f"
v== 4Tdfl ^ I tF (x + at sin 8 cos , y + at sin 8s\ncj),
z + at cos 6) sin 8d8.
Finally, the value of u is u = v + w.
chapter x. §§ 270, 274 173
-E^r. 4. As «, /3, 7 are constants, change the axes so that the p. 528
new axis of z is perpendicular to the old plane ax + By + yz = 0.
Then, in the new system, we have
a.r -t- By + yz = pZ = pR COS 6 ;
and the integral is
R 2 ( "d r e P Saos "sin 6 d6d(j),
Jo Jo
r>
the value of which is 4 _ u>
the result of the operation is to give
JJudS.
On the other hand, p 2 = V 2 , and so
4tt - sinh (RB) = 4nrR?
p
\ R 2 , R*
1+ 8l^ + 6T^+-
hence, on account of the equations
phi = 0, p*u = 0, ...,
we have JJudS = 4>ttR 2 Uo,
which establishes the theorem (due to Gauss).
§ 274. Ex. 1. Taking r and 6 as the polar coordinates of the p. 532
point x, y, we have the equation in the form
2 d 2 u du, d 2 u _
r dr i+r dr + d^~
The primitive is given by
u = A+B0+C\ogr
+ % {(A 1 r n + A 2 r- n ) cos nd + (B 1 r" + B 2 r~ n ) sin nO}.
Ex. 2. When polar coordinates are used as in § 271, the
differential equation becomes
r 2 dH d f Ju\ , 3 (, , , r,i) f I '<)"■„
a?d¥
dr\ T dr) + di*\ K 'M l-^d 2
174 CHAPTER X. § 274
Take u = e akH v ;
then v satisfies the equation
First, take v independent of <£ ; having regard to the form in which
derivatives of v with respect to p. occur in the equation, substitute
v = 2P n R n ,
where P n is the Legendre function of order n, and R n is a function
of r alone. The equation for R n is
Jr ( r ' IF*) ~ n (W + 1} ^ + ^^ = °'
the primitive of which is
Rn = i {^„e-*-/„ (»*r) + J?„^/„ (- ifcr)},
where ^4 rl and _B n are arbitrary constants varying from one function
R n to another.
p. 533 For the more general solution, we take v dependent upon
and expansible in a Fourier series
v = 2 w m cos (mcj) + a) ;
then w m satisfies the equation
Again, having regard to the form in which the derivatives of w m
with respect to //, in this equation and noting the result of Chap. V,
Misc. Ex., Ex. 12 (p. 64, ante), we substitute
M =
where P mn = (1 - ^ m ^ ;
and then R n satisfies the equation
& ('" If) ~ " {U + 1} Rn + kV ' Rn = °'
the same equation for R n as before. Thus the solution is
u = e aUi 2tP mn cos (m + a) R n .
chapter x. §§ 274, 278 175
Ex. 3. Substitute u = v cos akt ; then the equation for v is
9r 2 ?■ 3r r 2 90 2
Taking v=-R n cos (w# + a),
the quantity R n satisfies the equation
dr 2 r c?r \ ?
whence the result.
§ 278. Ex. 3. Proceeding as in the text, we have p. 541
p - q'y' = }
2/' 2 -P=±<7.y'J
as the subsidiary equations. Thus
'tyy"-q'y'=±(qy" + q'y')-
Taking the lower sign, we have
y" = o,
so that y = or,
y =ax + ' (a) da.
The quadratures can be effected by taking
ft, = a"" (a).
Ex. 4. The original equation can be resolved into the two
equations
r - \t {2p + f + q (4p + qrf} = 0,
r - %t {2p + f - q (4p + g 2 )*} = 0.
Having regard to the foregoing equations in Ex. 3, viz.
a 2 + aq—p = 0, y = ax + + q 2 f = {(4 being any function ; with
CHAPTER X. § 278 177
the positive sign of the radical, it is an intermediate integral of
the second resolved equation. Thus
2y + qx-x (4>p + qrf = {(4>p + tff - q],
2y+qx + x (4p + q 2 )% = -\fr {- (4>p + g 2 )* - q).
The integrals are obtainable as in the preceding example, by
taking (as there)
Ex 5. The subsidiary equations are p. 542
9 (£>' -?'' (a) +t'(/3)
y = «<£' ( a) _ «£ („) + py (£) - yfr (0)j
while (3 = z.
Ex. 6. (i) Integrals of the subsidiary systems are given by
qy = a, p + x = /3.
The primitive is
a=J-G'(/9)
,}
J-,
/3
^ + ^ = a + ^ (a) - a*" (a) + G (/3) - /3G' (£)
of which another form is
\x* + z = x/3 - * (£) + © (£))
a; = i '(£)- 8' (0)
the second equation in the latter form is the derivative of the first
with respect to /3.
f. 12
178 chapter x. §§ 278, 280
(ii) The integral of this equation has already been obtained,
§ 250, Ex. 5, (iii), p. 158, ante.
(iii) Two independent (and simultaneous) first integrals are
x 2 + y 2 + (px + qy) 2 = F (px + qy — z)\
py - q® = G (y\ \ •
y (1 + p 2 + q 2 f w J
The primitive is given by the elimination of £, t], t, between
£ = px + qy-z, v = ¥, ^=p + q^,
£ + (l + P + 7? 2 )* = 0?)^(4
p. 549 § 280. Ex. 2.
(i) z = xF' (x + y)-F(x-\- y) + xG' (x -y)-G(x + y);
(ii) There is an intermediate integral x 2 p — q 2 = 2a ; there
is an integral of the equation given by
^ = ^-J(2« + /3 2 ) + r
For the generalisation, take 7 a function of a and 0, and use the
method of § 280, finding the relation
d 2 " (/3) = 0.]
(iii) There is an intermediate integral (x + q) (y + p) = a ;
there is an integral of the equation given by
z = Py + p * - »'y + 7-
CHAPTER X. § 280 179
For the generalisation, take 7 a function of a and /3, and proceed
as in the last example, finding the relation
Take new variables a' and /3', given by
e 3a ' = a, e"' = /3« - -;
then the equation becomes ^7^ — ^- f . The primitive is
f GO
= - z + yer-'+P + xe a '~ * -xy\ = *-i^-i(:3a + /3 + 2y) ; r--|-(a-/3-22/) 2 -7 = )
3$ 33>
together with 7^— = 0, ^--pr = 0,
da 3/3
7 being determined by the equation
(vi) The primitive is given by
3 3
together with ^r— = 0. -^r = 0,
3a 3/8
7 being determined by the equation
The determination of 7 as the sum of two definite integrals can,
as in the preceding example (iv), be made to depend upon the
solution of an ordinary linear differential equation of the second
order which happens to be the equation of the hypergeometric
function F{\, —\, 1, x).
(vii) The primitive is given by
* = £-&(a + /3)«+:Ka-/3) 2 tf + (a-/3)y-7 = 0,
3 3
together with ^- = 0, ^ = 0,
while 7 is determined by the equation
, 3 2 7 = dy dy
3a3/3 3a + 3/8"
The linear equation upon which, as in the preceding example
(iv), the determination of 7 depends, is
d*G dG 1 ., n
CHAPTER X. § 280 181
(viii) The most general integral, expressible in finite terms,
appears to be
z = l(x + ay)+F(x),
where a is an arbitrary constant.
Another way of generalising the integral
z = %(x + ay) + ,5
is to make /3 an arbitrary function of x and a, subject to the
requirement that the equation should be satisfied. When x and a
are made the independent variables, the equation for /3 is
a/3 | d/3\ d*/3 = i ,
dx da) dxda '
but this is no easier to solve than the original equation.
Miscellaneous Examples at end of Chapter X.
Ex. 1. An intermediate integral of the equation is p. 549
+ y)(p-q) + %z =f{y - «)>
and this equation is linear of the first order.
Using Lagrange's method of integration, two integrals of the
subsidiary system are
Jv r _??
x + y = a, aze "■ — I e a f(Zy — a)dy = /3.
Take /3= — F(a), and substitute x + y = a; the result follows.
(See § 276.)
The second equation is changed into the first by the dual
transformation of § 253 ; and therefore its primitive is given by
the equations
® = (X + Y)Z + eX+*F(X+Y)-eX T r\e~^f(2Y-a)dY=0,
putting a = X + Y after integration, together with
x dZ + dX~°' y dZ + dY~°'
d~ .
cc
Ex. 6. Every integral of the equation py — qx = is of the form
x- + y* = 6 (z) or, say,
(* 2 + 2/ 2 )*=/(4
When substitution takes place in the equation of the second order,
the function / satisfies the equation
1 dz* \dz) V -
The primitive of this is
f= c cosh ,
J c
where a and c are arbitrary constants; hence a solution of the
original equation is
(x 2 + y 2 )- = ccosh-.
(The equation of the second order requires the associated
surface to be a minimal surface ; the equation of the first order
requires the surface to be one of revolution. The only surfaces,
satisfying both conditions, are catenoids given by the foregoing
equations.)
Ex.1, (i) z = F(e?+ey) + G(e x -ey);
(ii) y = F(x)G(z);
M ._„(£) + ./ ®*
,x
where, after integration, — is substituted for t
(iv) z-x*=F(y) + G(^
.2 1 >
184 CHAPTER X. MISCELLANEOUS EXAMPLES
(v) z= F {y + 2x^) + G {y -2x*);
(vi) z = axF' {ax + y) — F {ax + y)
+ axG' {ax — y) — G {ax — y).
Ex. 8. The second of the equations is satisfied by taking
du a du . a
7T- = cos a, sr- = sin t),
ox oy
and then — sin 6 ^- = cos 6 ^- .
3?/ o*
The first of the equations is satisfied by the assumed values for
du , du .,
* - sin 6 ■=- + cos ^- = 0.
8a; 8?/
r)/9 pi/3
Hence ^- = 0, ^- = 0, so that 6 = a, where a is a constant ; and then,
dx By
by quadratures,
u = x cos a + y sin a + /3.
p. 551 Ex. 9. Let a 2 + b 2 = c 2 . The two equations are satisfied by
taking
r = a + c cos 8, s = c sin #, t = a — c cos 0,
where # is a new variable. Now
Br _ds ds _dt _
3y 3* ' 9y 3* '
, ■ Q dd a d6 a dd ■ aSO
so that — sin u ^- = cos ^— . cos — = sin ^- .
83/ oa; ' dy ox
Hence — - = 0, ^- = 0,
3a; 83/
so that = a; thus
r = a + c cos a, s = c sin a, t= a — c cos a,
and the result follows at once by quadratures.
Ex. 10. One intermediate integral is
q{i + P r i = G'(y)-
The subsidiary equations are
dp _ dq
-Q-^Wty)--'
so we take p = a.
CHAPTER X. MISCELLANEOUS EXAMPLES
185
Then q = (1 + a?f G' (y),
and z = ax + (l + a 2 )* G (y) + b.
The general integral is given by
z = ax + (1 + orf- G(y) + $ (a)
=x + a (I + a*y$Q(y) + $ (a))
Another intermediate integral is given by
pz + x = F(p);
and a primitive is obtained by eliminating p between this
equation and
z (1 +p*f =Jp (1 + p 2 )~iF' (p) dp + ay + b.
The general integral is obtained by taking b = (f> (a), with
V + f O) ;
that is, we eliminate p between
pz + x = F(p) \
z(l + f)i=f P x F'(p)dp + G(y)\ '
■/(1+p 8 ) - - j
Ex.ll.&z = F^) + G(l
(ii) This equation arises (i) by the dual transformation in
j 253 ; the primitive comes from the elimination of X, Y, Z between
z = xX + yY — Z
(X 2
' = f {t) + G K:,
x = 2
rft
Y
P
I
F 2
Y* (Y*
X* G \X,
y =-~FT-^) + 2^G'
X-
Y
Y
'X
(iii) z = (x + y)F(x-y) + (x-y)G(x + y);
(iv) z = F (xy) + xG (xy) ;
fy\ j f(e^)d$
(v) z=F(xy) + xG(^)e-
where, after quadrature, { log (xy) is to be substituted for £.
186 CHAPTER X. MISCELLANEOUS EXAMPLES .
Ex. 12. (i) xz = F (y + ax) + G (y - ax) ;
(ii) 2x 2 z = xF' {x + ay)-F(x + ay)
+ xG' (x — ay) — G (x — ay) ;
<<-'(»0 +e (»-i)'
(iv) z = F'(y-ax)-G'{y) + ~ {F(y-ax) + G(y)};
LIJi/
(v) (x + yjz = (x+y) [F 1 (x) + G' \y)} - 2F(x) - 2G (y).
^*. 13. The equation of the surface is
x(y + ^) = a 2 .
p. 552 Ex. 14. (i) u + iv = F(x + yi) + G(x- yi),
where i denotes *J— 1, and F and G are to be resolved into their
real and imaginary parts ;
(ii) When m and n are unequal, the only common integrals
are a = and /3 = 0. When m and n are equal,
tx= Yx^ x,y ^ P = tyf( x >y)'
where / is any function of x and y.
Ex. 15. The complete integral of the second equation is
Substitute this value of z in the first- equation; then
cc'-
A = -
c+c"
so that the integral relation is
n cc ' 2 cc ' 2
z — 15 = ; # 2 -, y",
c-c c+c"
from which the geometrical interpretation follows at once.
Ex. 16. The equation can be written
Gdq + Hdz + Kdx = 0,
with the assumption that y remains constant. When the given
condition (of integrability, § 151) is satisfied, the equation has a
single integral equivalent ; the constant of integration, the process
of which follows the process of § 152, must be made an arbitrary
function of y.
CHAPTER X. MISCELLANEOUS EXAMPLES 187
As regards the example, the intermediate integral is
{x + y)q${y) = yz+x.
Regard this as a linear equation in z, where y is the independent
variable and x is parametric ; integrate this linear equation, and
take the arbitrary element after the integration as ^r{x). The
result follows.
Ex. 17. Write Jl + j^ = <9;
oy 2 oz-
the required integral is
u = (l-p>+f ] 6<-...)F(y,z)
+ (v-^P+f,f--)G(y,z).
As regards the second equation, it is always possible to transform
(a,b, c,f,g,h\%, v, O 2
to the form f 2 -f rf + f 2 unless
= 0;
a,
h,
9
h,
b,
f
9'
f.
c
so, save for this exception, the equation becomes
/ 3 2 3 2 d 2 \
fe + SY 2 + 87 2 J M = '
the integral of which has just been given.
In the exceptional case, the equation is equivalent to two
linear equations
du „du du .
a Tx + ^Ty + ^z^
the integral of which is
u = F (#/3 — ya, xy — za).
Ex. 18. Let x = r~j y- 1+b + ax, y' = — ^ y~ 1+b - ax ;
the equation becomes
d*z c fdz_ dz \ _
d7dy~' + ^+? \dx + dy'J ~ '
i b
where c = £ , _ .
188
CHAPTER X. MISCELLANEOUS EXAMPLES
Forming the quantities K of § 256 for the successive trans-
formations of this equation, we have
(a
' + yj'
(1
+ c)(2-c)
(x' + y') 2
(*'
+ c) (i + 1 -
c)
Ki (x' + y'Y
The equation is integrable in finite terms (§§ 256, 257) if any K
vanishes. This happens if c is a whole number, positive or negative,
that is, if
b(2i±l)=2i,
where i is a positive integer.
p. 553 The primitive of the second equation is
x dx
{F(y + ax) + G(y- ax)}
when i is a positive integer, and is
,/l 3
{F(y + ax) + G(y -ax)}
\x dx)
when i is a negative integer.
Ex. 19. Substitute u = vjr ; the equation for v is
d 2 v 1 &v_ u(n+l)
dr 2 a 2 dt 2 ~ r 2 V '
the primitive of which occurs in the previous example. The
required value of u is deduced at once.
For the second equation, take
r + at = x, r — at = y ;
the new equation is
2^ + *
7 (^ + f^ = 0.
Next, take
then
dx dy' ' x + y' \dx ' dy
u = v (x' + y')~* ;
. d 2 v
dx'dy' (x'+y'y
= 0.
CHAPTER X. MISCELLANEOUS EXAMPLES 189
Two integrals of this equation are
where F T and G a are arbitrary, and © (<) is the hypergeometric
function
F(h i, 1, t),
that is, © (t) is the first elliptic integral having t for its modulus.
The primitive follows.
Ex. 20. There is a misprint in the text ; the equation should be
d'V = 1 /gF dV\
dxdy x — y\dx dy J '
It is deduced from the preceding equation by taking
x = x, y = -y;
and the primitive follows.
Ex. 21. The method of § 270 can be extended. Changing the
constant I, we have
J — CO
Also ir*P*-f e-^ + ^^du;
J -CO
thus • W=f f e-^-^+^^dudo.
J -co J -co
Symbolically, we have
and therefore, taking I to be the symbolical operator at*K-, we
have
ttm' = I e dx (p(x) dudv
J —aoJ -co
= f " ( e"" 2 -" 2 (a; + 2 f aw^ <*) cfecfo.
J — 00 J —CO
where /( * )= /^
190 CHAPTER X. MISCELLANEOUS EXAMPLES
Ex. 22. When the transformation of variables is effected, the
new equation can be taken in the form
dx 2 dzdx
OX oz
so that a first integral is
The primitive of this equation of the first order is
x+f(z)=F(x + y + z),
dz
Ex. 23. Take five functions Z lt Z 2 , Z s , Z it Z t , such that
Z= z 1 Z 1 + ... +z 5 Z 5 ,
P= Pl Z 1 + ...+p 5 Z s ,
Q = q 1 Z 1 +...+q 5 Z s ,
S= s^ + .-.+s^;
and choose the functions so that Z, P, Q, S vanish. Forming dZ,
dP, dQ, dS, and using the assigned equations, we have
z 1 dZ 1 + ... + z 5 dZ 5 = 0,
p l dZ 1 + ... +p 5 dZ 5 = 0,
q,dZ,+ ... + q,dZ, = Q,
s.dZ^ ... +s 5 dZ 5 = 0.
Taken in conjunction with Z = 0, P = 0, Q = 0, 8 = 0, these
equations shew that
dZ ± _dZ 1 _ _dZ 1
Zi Z 2 Zc,
Thus the ratios Z x : Z 2 : Z 3 : Z± : Z 5 are constant ; when they are
inserted in the equation Z = 0, they give
C 1 z 1 +... + C s z s = 0,
where the quantities C are constant.
p. 554 When
Z \, Z 2, ■%> Z k ; ")
Pi, Pz, Ps, Pi
qu 92, q*, q<
5j, 5 2 , S3, S 4 i
the equations Z=0, P=0, Q=0, S=0, giveZ 6 = 0, that is, C 5 = 0;
whence the second result follows.
CHAPTER X. MISCELLANEOUS EXAMPLES 191
Ex. 24. Denote the coefficient of ac m y n in F by G m>n ; then
(m + n + a) (m + /3) C m ,» - (m + 1) (m + 6>) C m+hn = 0,
O + m + a) (n + y) C m>m - (n + 1) (n + e) Mi „ + , = 0.
Write
S =
: *• - , S = y -
dx' y dy'
these relations between the coefficients C shew that F satisfies
the equations
(* + &' + a) (* + £)--*(* + - 1)1 y = 0,
(* + &' + «)(*'+ 7 ) -*y(a'+e-lHy = 0,
which are the given equations.
Add the two equations, so that F satisfies their sum ; take
ft = 8+c, 7=-c;
then F (a, 8 + c, — c, 0, e, x, y) satisfies the final equation.
Ex. 25. The process is similar to that in the preceding
Example 23. Four functions Z lt Z 2 , Z s , Z A are chosen so that
Z= z x Z x + z 2 Z 2 + z 3 Z s + z 4 Z, = 0)
P = p 1 Z 1 +p 2 Z 2 +p 3 Z 3 + Pi Z t = {)[;
Q = q 1 Z 1 + q 2 Z 2 + q 3 Z s + q,Z, = oj
then z 1 dZi + z 2 dZ 2 + z s dZ 3 + e t dZ 4 = 0|
p x dZ x + p 2 dZ 2 + p 3 dZ 3 +pidZi = 0> -
q l dZ 1 + q 2 dZ 2 + q s dZ s + q^Z^ = OJ
Taken in conjunction with the earlier equations, these shew that
dZ l dZ 2 _dZ 3 _dZ t
Z 1 Z^ Z 3 Z^
so that the ratios Z l : Z 2 : Z 3 : Z± are constant ; thus
C x z x + C 2 z 2 + C s z s + C 4 4 = 0.
But, if
Pi, P*> Ps
while Z = 0, P = 0, Q = 0, then Z t = 0. The result follows.
0,
192 CHAPTEE X. MISCELLANEOUS EXAMPLES
Ex. 26. Differentiate the first equation n times with respect to
x, and write ^n 2
dx~ n = Z "'
the equation satisfied by z is
s + xyp' + (k + n) yz = 0.
When k is a negative integer, take n = — k ; the new equation is
s' + xyp' = 0,
and its primitive is
s ' = F 1 (y)+ \ X e-l x y*G(x)dx.
This integral contains two arbitrary functions ; consequently, in
deducing z by ra-fold integration with respect to x, there is no
necessity for adding arbitrary functions of y in the process. The
primitive is rrx „
z = x n F(y)+ Jl ...e-^ x y i G{x){dx) n +\
When k is a positive integer, differentiate the equation
s ! + xyp =
k times with respect to y, and interchange x and y ; then
* = *(y) + ^jy iyx *G(y)dy,
p. 555 Ex. 27. (i) The verification is immediate.
(ii) In the first equation, take e z = & ', so that it is
dxdy '
and write z = ^- ;
then, integrating with respect to x, we have
Taking F (y) = 0, we have S = ZP ; whence the integral follows.
(iii) Let (x) dx = x, $ (y) dy = y'; the equation becomes
d a z z
dx'dy'
The result follows from using the given integral of the preceding
example (i).
CHAPTER X. MISCELLANEOUS EXAMPLES 193
Ex. 28. (i) The primitive is
together with r^=0.
(ii) An integral is
z — ax + (3y — yxy.
Adopting the process of Imschenetsky's generalisation (§ 280), the
equation to determine 7 is
8a 2 8/3 2 da ti/3
which is a linear equation with constant coefficients to be integrated
by the method of § 262.
(iii) The primitive is given by the three equations
= - z + ax - \a? + /3 log y + /3 log (- log a) + F (a) + G (/3) = 0,
oa dp
f. 13
GENERAL EXAMPLES OF DIFFERENTIAL
EQUATIONS.
p. 558 Ex. I. (i) Primitive* is 4y = (x + Af.
Singular Solution f y = 0.
(ii) Take px — y = Y; then
dY l-2p 2 ,
-Y= dx - 3 { l- p){ l-2p-2f) d P-
Integrate, and use § 30.
(iii) There is a misprint ; the equation should be
p (ny — px) = c.
Pr. given by associating the equation with
No S. S. unless n = 1, in which case the primitive fails ; the equa-
tion is then of Clairaut's form.
(iv) Pr. given by associating the equation with
1 + p 2 i r 1 1 + p 2
No S. S.; for the ^-discriminant leads to y — x — 1 = 0, which does
not satisfy the equation. (The student should make out the
significance of the locus y = x + 1.)
(v) Pr. is 2x 2 -2y 2 -a 2 =(2x + A ) 2 .
S. S. is 2x 2 - 2f - a 2 = 0.
(vi) Pr. is (x + y + A) 2 = 2xy, or >Jx + *Jy = *J A.
S. S. given by x = 0, y = 0.
(vii) Pr. given by associating the equation with
X = A P + -*-
No S. S.
* Pr. will often be used for primitive.
t S. S. will often be used for singular solution.
GENERAL EXAMPLES OF DIFFERENTIAL EQUATIONS 195
(viii) Pr. is y — §#* - 2ax% = A.
No's.S.
(ix) Take x = e e , y = ^e 28 , az — 1 = w 2 : then
u d + u — $a
(x) Associate x = Ap* + \cp~ 2 with the given equation.
(xi) Pr. is y = -do 2 + A 3 x.
S. S. is 2/ 2 +^V« B = 0.
(xii) Pr. is (x — yY = x + Ax 2 .
S. S. is x — y.
(xiii) Substitute y = Y 2 . The equation is
2/ = 8P7(4P 2 -l);
use § 18.
(xiv) Use the transformation a? = X, y 3 = Y, Pr. is
a , cA
S. S. is (x 3 -y 3 ) 2 -2c(x 3 + y s )+c 2 =0.
(xv) The equation gives
xp
p 2 -l
use § 18.
(xvi) Pr. is (y - Ax 2 ) 2 = 8Ax\
No S. S.
(xvii) Substitute x = rcos0, y = r sin 0. Pr. is
No S. S.
(xviii) Resolve for p : substitute x = e", y = •ze 49 . Then
50
= 4z + z 2 + z(4z + z 2 )i;
evaluate by the substitution z + 4 = 5W 2 .
(xix) Combine the equation with
(p-Yfx = A-p 3 +§p\
(xx) Pr. is y 2 = 4>A(x — A), the equation of confocal and
coaxial parabolas. There is no S. S. that is real.
13—2
196 GENERAL EXAMPLES OF
(xxi) Pr. is (y — A) 2 + x 2 = A 2 sin 2 a, where c = sin 2 a. S. S.
is y 2 = x 2 cot 2 a.
(xxii) Take Y = y 2 , X = x 2 ; the equation is
F - pz = TTP 2 -
A
Pr. is y 2 -Ax 2 = -, —
y 1 +
S.S.
is
A 2
4 {Sy 4 - (x? + l) 2 } f Sx 2 (x 2 + 1) - y A ] = y 4 (1 - 8« 2 ) 2 .
(xxiii) « 2 (j/ 2 — 2xy — x 2 ) = A.
_i
(xxiv) Combine the equation with xp 2 e p =A. There is
no S. S.
(xxv) Substitute x = e e , y = ze^ ; Pr. is xy (x 2 — y 2 ) = A.
dz
(xxvi) Substitute x = e e , y= ze~ 2e ; then \ -^ = u, where
(m-2) 2 =^ 3 (3-m) 3 .
(xxvii) Substitute x 2 — y = z. Pr. is
—!—e-i xi =A-[xe-l xi dx.
x 2 -y J
(xxviii) — x = A(\ -x 2 v.
y
p. 559 (xxix) y 2 — x 2 = Ax.
(xxx) Combine the equation with
x + A '= fp 2 + log p.
(xxxi) A Riccati equation (§§ 108-9) for which
6=1, c= — a 2 , m = — 4, i = 1 ;
the equation is integrable in finite terms.
(xxxii) — — - = Ae~ ax - a 2 x 2 + 2ax - 2.
xy +1
(xxxiii) Homogeneous equation. Pr. is
y 2 - x 2 = A (y - 2a;) 2 .
(xxxiv) Substitute y = ux, u = v + l. Pr. is
(2y + x\ 2
A + log(y-x)=^ (
(xxxv) (x — yf = Ax (x — 2y — 4).
(xxxvi) x 2 y + iy s + a 2 y = A.
DIFFERENTIAL EQUATIONS 197
(xxxvii) Pr. is A 2 - 2Ay~* + 4>x = 0. There are two singular
solutions, viz.
y = 0, 4:xy 2 +1=0.
(xxxviii) Pr. is f = 3 Ax 2 - A 2 . S. S. is 4 - {sinh £ (0 - 0)}">+ 6 = A .
(xlii) Pr. is 4a* -y 2 = (A + x) 2 . S. S. is y 2 = 4ax.
(xliii) The equation should be
y(x—y — xpf —p (x 2 — 2xy) (x - y — xp) 2 + yp 2 = 0.
The Pr. of this equation is
x 2 — 2xy = Ay 2 + -j .
S. S. isx 2 -2xy = ±2y.
The equation, as printed, appears not to be integrable in finite
terms.
(xliv) Substitute y = PX - Y, p = X, x = P; then
-1 1
Y 2 = AXe x - j+2.
Proceed as in § 30.
(xlv) y 2 -2A(y + 2x) - 3A 2 = 0.
(xlvi) Eliminate p between the given equation and
2
y — x = Ae p_1 .
(xlvii) The equation, as printed, appears not to be integrable
in finite terms. When the substitutions x=rcos0, y = r sin 0,
are used, the equation becomes
r 3 ~ cos 20 + a 2 (^ sin 20 + r cos 26>Y = ;
but apparently there is no simple integral equivalent.
198 GENERAL EXAMPLES OF
When the similar equation
xy (x> - y") (p 2 - 1) + (x 2 - y 2 ) 2 p + a? (ocp -yf=0
is propounded, the integral can be expressed in finite terms. The
equation can be written
(a? - y 2 ) (yp + x) (xp - y) + a 2 (xp - yf = 0.
Corresponding to the factor equation xp — y = 0, there is an integral
y = Ax.
The Pr. of the remainder is
r 2 + a 2 log (tan 20 + sec 20) = A ,
where x = r cos 0, y = r sin 0.
(xlviii) Resolve for p, and substitute Z = (a 2 — &)y 2 — cV;
then {c 2 (a 2 -c 2 )-£}*+2aa; = A.
(xlix) Substitute a; = r cos 0,y = rsin0,K = (4a 2 — l) 3 ; Pr. is
r = ^e" 9 -
(1) Pr. is x 2 = Ay'" + A-+ 1.
S. S. is 2/ 6 +4* 2 =4.
i?*. 2. The differential equation is formed by eliminating c
between the given equation and its first derivative ; it is
4# 2 (y — ax) = 2a? (p — a) — 6 2 (p — a) 2 .
The ^-discriminant gives
4& 2 (y — ax) = of,
which is a singular solution.
The c-discriminant gives the preceding singular solution ; it also
gives y — ax = which is a particular solution, corresponding to
the value c = a.
dA dA
Ex. 3. Writing ^— = A 1} j— = A 2 , and so for B, we have
c (A, +pA 2 ) + 2 (B 1 + pB 2 ) = 0.
Eliminate c; then
L=A 2 - 4B 2 ) (A^B., - A..B,) 2 ,
the required result.
DIFFERENTIAL EQUATIONS 199
When the integral curves have an envelope, it is included in
A - B 2 = and therefore in LN - M 2 = 0. Similarly when they
have a cusp-locus.
When there is a node-locus, at every point of it there are two
values of p for the integral curves ; hence
cA 1 + 2B 1 = 0, cA 2 + 2B 2 = 0,
and therefore T= 0. Hence the node-locus is included in
LX-M 2 = 0, but not in A - B 2 = 0.
Ex. 4. Taking logarithmic differentials, we have
dy _ 2xdx
y 1 — x 4 '
ai t i /l-« 2 \ 2 4a; 2
Also 1 - y* = 1 - = ■
9 Vi+W (i + * 2 ) 3 '
therefore ^ l + x*f_2xydx^
(1 _ yiyi 2x \ \-x*
— y—dm
1 — x
dx
(1 - a;*)* '
Ex. 5. (Compare Misc. Ex., ch. viii, Ex. 5, p. 371 in the text; p. 560
a general method of solution is given p. 126, ante.)
Take the curve rf = 1 + f 3 , and the line 77 = mf + c. They cut
in three points; let the abscissae be x lt x 2 , —a. Keep a fixed;
simultaneous changes in x x and x 2 will change m and c. Now let
/(f) = f 3 + 1 - (wf + c) 2 = (f - O (f - * 2 ) (| + a).
™ -4f + £ v ^X+5 1
lhen — .. „. = 2,
/(f) " /'(Z) f-X'
where the summation is over the three roots of /(f) = ; thus,
taking the coefficient of -r on both sides, we have
Now 2rjdy = S^ 2 d^, dr) = md^ + %dm+ dc,
for simultaneous variations ; hence
d% _drj _ gdm + dc _ %dm + dc
2^~W~Z¥-2mv~ /'(f) '
200 GENERAL EXAMPLES OF
Taking A = dm, B = dc, we have
dx x dx s da _ ~
2^ + 2^~2(l + a'7 2 ~ ;
or, as a is kept constant, we have
+ _ 2 =
2/1 2/2
as the differential relation. Further
«! + # 2 — a = m 2 , ^#2 — &#i - gm?2 = — 2a??i, aa?^ = 1 — c 2 ;
and therefore
(aj^a — 0^1 ~ ax 2 ) 2 = 4 (*! + * 2 — a) (1 — axxX^),
is the equivalent integral relation. It is the required integral,
with a as an arbitrary constant.
Ex. 6. Taking y J + M' + y") we find
5 y+\(a> + y-b)
' dY -xf 1
F rfZ V* - a a - 6/ '
whence the result.
When \ = 0, Pr. is
. 1 1 . os — b
A + - = r log .
y a—b ° x—a
When\ = -1, Pr. is
y . 1 . x — b
/ V7 n = A + — T lo g ■
(x—a)(x—b) a- b n x — a
Ex. 7. Take/(#) = — 1. The equation for z is
(x - a) (x - 6) 2" - {(« - a) + (0 - 6)} 3' + 2s = 0.
Pr. of this equation is z = B (x — df + G (x — b) 2 ; hence
_ 2z _ 4 Q - a) 2 + (a; - 6) 2
2/_ #' " 4 (# - a) + « - 6 '
,j. 2y + # — a 1 « — 6
leading to -^ =- = — ;
Zy + x-b Ax- a'
in agreement with the preceding example for X = 1.
When a = b, Pr. is
_ _ _ A (x — of + {x - a)
V ~ 24 (x - a) + 1 '
DIFFERENTIAL EQUATIONS 201
Ex. 8. The general equation of the first order is F(x, y, p) = 0.
Resolve for y, and let the result be
y = px+f{x,p).
The primitive is to be
y=ax+f(x, a).
df(x a)
But p = a ; so • , ' — = ; i.e. f(x, a), and therefore f(x, p), does
not involve x. Accordingly, the equation is
y=px+f(p).
Ex. 9. By hypothesis, the equations f(x, y, p) = and
(f> (x, y, c) = coexist. Apply to both of them the transformations
of § 30 ; then the equations
f(p, xy -p, x) = 0, (p, xy -p,c) =
coexist. Hence the result.
Ex. 10. By reciprocation with regard to y 1 = 2x, the two
equations
v-y=p(£- x )> Ft?-|-Z = 0,
must be the same (§ 30, Note) ; hence
F =i, y—x=x.
p p
Thus dY=--dp, dX = ^--^- 2 dp-dx = -^dp,
p r> p p p r
so that P = - . Hence, when the equations
y
f(x,y,p) = 0, (x,y, c) =
coexist, the equations
f(y—x, 1 -, i)=o, *(*—, i, c )=o
\p p yi \f p i
coexist ; whence the result.
For equations o.f the second order, let q = d 2 y/dx>, Q = d 2 Y/dX*.
From Py = 1, we have ydP + Pdy = 0, i.e. yQdX + Ppdx = 0, or
,,3 P3
p
P
202 GENERAL EXAMPLES OF
Hence, when the primitive of f(x, y, p, q) = is known, we can
deduce the primitive of
f(V. m , I, I, JQ=o.
\p p y qp 3 /
Pr. of given equation is
ye -i x2 =A+BJe- x2 dx.
Ex. 11. (i) y = Ae x + Be™ + ^e -2 * + \ xe ™ \
(ii) y = Bx + A(l + x 2 )i - Ax log {x + (1 + x^} ;
(iii) y = Ax + Bx^ cos fa + log ^j^J + «s log x + \x 2 + ^%x s ;
p. 561 (iv) y = (A + Bx) e~ x
+ e$ x {(A' + B'x) cos (x V3) + (A"+ B"x) sin (* V3)}
(v) l g_^ = 5 + -;
(vi) y = 4 + - + — ;
(vii) t/e-^-^^^ + SJe- 2 ^-^ 3 ^*';
(viii) y = x~% \A — i / x~isinxdx>
is an
#? — CO
imaginary cube root of unity ;
(xiv) y=Ae- x + B(x 2 + x + S);
(xv) y = x" (A cos ^ + B sin £j ;
DIFFERENTIAL EQUATIONS 203
(xvi) y = A cos (e~ x ) + B sin (e~ x ) ;
(xvii) y = A cos (sin «) + 5 sin (sin #) ;
(xviii) yei* 2 = Ax~ 6 + Bx* - \ a? + -^ ;
(xix) Take z = x + 1 ; then
s 2 + 2 U 2 + 2 V2 V2j '
(xx) See second part of Ex. 10, p. 202, ante.
(xxi) Taking ^ = x -=- and proceeding as in § 114, the equation
can be written
{x*(*>-2)(*t+2)(* + 3) + %(*-l)(<$ + l)}y = 0.
Let y = Boas"- + B^xf--*-* + . . . ;
the critical values of ft are given by
/* 0* - 1) 0* + 1) = o,
and the relation between the coefficients 5 is
-B 2 »-2 (/* + 2w - 4) (p + 2n) (ji + 2n + 1)
= - B 2n (fi + 2n) (fj, + 2n -l)( l i + 2n + 1).
The value ji = gives
y = A(l + 2x 2 ).
The value /j, = 1 gives
y + x (1 + a; 2 )^ = Bx.
For the third special integral, belonging to /j, = — 1, proceed as in
§77.
Or Use the Frobenius method, in the Supplementary Note I
at the end of Chapter vi.
(xxii) A particular solution is y = x ; use § 58 : Pr. is
l-fx*el x °'dx = A+B[x->el xl - x dx.
x J J
(xxiii) Multiply by cos x ; the equation is
y" cos x — y' sin x + 3 (y sinx + y cos x) = tan 2 a;,
leading to
r, a ,, i , , x fsin x — x cos x 7
v sec 3 x = B + A (tan x + * tan 3 x) + ax.
17 \ s > j cos 5 «
(The last integral should be evaluated.)
(xxiv) yet* 2 = Ae x + Be~ x ;
204 GENERAL EXAMPLES OF
(xxv) A particular solution is y-x, when the right-hand
side is zero ; use the method of § 58 ; Pr. is
= B + A T j 2 e~i zi dz + T j 2 e~i zi dz T ei ui (ulog u) du;
p. 562 (xxvi) The equation should be
(4a! . _ ^ _ 9s) g + (6^ - y 2 ) g + n*y = 0.
Take a new variable du, given by
die = (4* 3 — g 2 x — g 3 )~ 2 dee.
[In effect, x = <@ (u), the Weierstrass elliptic function.] Then
y = A cos nu + B sin nu.
(xxvii) A particular solution is y = x*l(x—l), obtained by
substituting y = x m (x — 1)", and determining m and n, so that the
equation is satisfied. Pr. is
y"-^± = B + A\2log ' ; ""' !
X 2 { x — 1 x (x — 1)
Or reduce to the canonical form (§ 60), which is
d*z 2
j~« 7 ^ z = °>
ax- co (x — 1)
of which z — x{x— 1) is a particular solution; use the method of
| 58, and obtain the foregoing result ;
(xxviii) Let u = xy; equation for u is
d' 2 u du , , . u .
the primitive of which is given for Ex. 19, Misc. Ex., ch. v (p. 66,
ante) ;
(xxix) Let F(n) denote
2« „ , 2 2 ?i(?i-l)
x 4- ■ '"
l(2n-2) 2!(2w-2)(2n-3)
Pr. is y = AF(n) + BF (- n - 1) ;
(xxx) y = x(A cos x + B sin *•) ;
(xxxi) ye~ x = A + B le~ ix2 (2x-l)~* dx;
DIFFERENTIAL EQUATIONS 205
(xxxii) Denote the series
x m + c 1 x m+i + ... + c p x m+i *> + ...,
where the constants c are connected by the relation
(m + 4p) (m + 4p — 1) (m + 4p — 2) Cj, = ac p - x
(c = 1) by F (m) ; the primitive is
ay = AF (0) + &F (1) + CF (2) ;
(xxxiii) xy = Ae™ + Be x ;
(xxxiv) A particular solution, obtained as in (xxvii) ante,
is (1 — x) (1 — 2x)^ ; use the method of § 58 ; Pr. is
— X ° 1 - X
(\-x)\\-2xf- W
(xxxv) l = B + A(l-^-l-±-\o g «
-2J^(x-2)log(x-2);
(xxxvi) Change the variable (as in § 63) by the relation
z" + Pz' = 0,
say z>J2=-\x * (a- — l)^dx;
Pr. is y- Ae z + Be~ z ;
(xxxvii) ye~^ x3 = A+B je ixi ~ ix3 dx;
(xxxviii) Let a and /3 be roots of 6 (6 — 1) + bd + c = 0;
Pr. is ye ax = Ae ax + Be fix .
(The case when a = /3 should be discussed.)
(xxxix) y = Ax + Bx 2 + \ — 2x log # + x 2 log # ;
(xl) xy = A(x-2)+ B(x + 2)e- x ;
(xli) y = -4# + B cosh -1 x ;
(xlii) A particular solution, obtained as in (xxvii) ante,
isy = x/(l+x). Substitute y = ux/(l + x) : then
/ 1 \ 1 Ac + 1
= A+B {~x +l0ga: J'~x + J^ 1 °g(^+ 1 )^-
tt :
206 GENERAL EXAMPLES OF
Ex. 12. y = x (A sin x + B cos x + Gx).
Ex. 13. y = e - * aV (A + Be ax + Ce~ ax ).
p. 563 Ex. 14. y = Ax + B sin x + cos x.
Ex. 15. y = (Ax + B) (sec «)*.
Ex. 16. Using the method of variation of parameters (§ 65),
we have
^,£ + ^^ = — d>' + — Tlr' = F
doc dx * ' dx ™ dx *
Resolve for -=— , -=— , and integrate ; the result follows.
Ex. 17. Let a = 2n, and write xy' — 2ny = u. The equation
for n is
u" - c 2 u + (2n - 2) y" = 0,
so that, when a = 2, m = -4e CT + Be~ cx ,
and therefore ^ - = G + \(Ae ex + Be~ cx ) — n .
x 1 ] X 2
For other values of n, we have
u'" - c 2 u' + (2k - 2) y'" = 0,
that is, u'" - cV + (2n - 2) c 2 - = 0.
x
Hence ^AfVA-ll^
or, writing x"-z= 1, we have
Let !T denote ^e" 2 "^ 5e- cz_J ; then
(- 2)» i*y u = JT (&)» + CU + C„_ 2 * + . . . + C^.
Substitute, and equate coefficients. Thus
2/ 2 = ^f/(^- +J B e -)J + ^(l-^J).
(XT/
i?#. 18. Write -j- = xu\ the equation for it is
xu" + (a + 2) v! + bxu = 0,
which proves the first result.
For the primitive, let a = 2n ; then
(1 d \"
-^J {4cos(«V&) + -Bsin(*V6)}-
DIFFERENTIAL EQUATIONS 207
Ex. 19. Take x = cosh.u, so that f = e u ; the equation in y
and u is
g + coth W |-n( M + l)y = 0;
so, if y = 7je _n ", the equation for r\ is
t-( — (2m — coth m) ~p — n (1 + coth u) i] = 0.
Now let 2 = e 2K =£ 2 ; the equation becomes
0(i_2) S +{ * _n+(?i - f) ^S + * n ' ?=() '
a hypergeometric equation for which
a = - n, £ = £, 7 = £ - n.
-£"«. 20. Let y = 2 cos m = e iM + e _iu ; the given equation is
cos 5w= 2% — 1,
so that the roots y are of the form
cos u, cos (u + |7r), cos (m + 1 7r), . . . ,
five expressions linearly expressible in terms of two quantities cos u
and sin u, so that the differential equation for y should be of the
second order. Now
5m = cos -1 (2a; — 1),
so that 5-r = -
dx (x - x*f
and therefore i=i sin u . r ;
dx (as- x*f
consequently
d?y 1 , . l-2x
, , = - A cos M . - 4 sin M
dx 2 5 a; — x 2 D
i/ . cfa/ 1 — 2a;
(a; - x 2 f
3.
,-*
K « — a; 2 2 dx x— x 2 '
leading to the given equation.
i?a;. 21. When the given equation possesses an integral f(xjm), p. 564
then
Lf» (£) + $ {x ) -f (-) + + (x)/ (-) = o,
so that /" (*) + m$ (nut)/' (z) + m 2 ^ (mz)f(z) = 0.
Similarly, from the integral f(x/n), we have
/"' (z) + n4> (nz)f (z) + n*f (nz)f(z) = 0.
208 GENERAL EXAMPLES OF
Hence « = -£& = m tM-«'f(") _
f(z) m (mz) — n (ms) y 1 + m 2 i|r (wis),
/ K z )
leading to the theorem.
There is an obvious failure when m = n. When tn — — n, the
numerator for y r is an odd function of z, and the denominator is
an even function oiz: thus /(a) is an even function of z, that is,
f(x/m) =/(- ccjn) =f(x/n),
aud so the given expression is not a primitive of the original
equation.
The primitive of the final equation is
y = A(l-l)ehB(l-l),
Ex. 22. Take z = (1 — 2*) 2 ; the equation becomes
*(i-*)g+{i-G« + i/3-i)*}^-W^ = >
a hypergeometric equation for which |a, £/3, ^ are the elements.
The primitive follows from § 115.
Ex. 23. For the first part, substitute x = — in the equation
z
of the hypergeometric series for the given function. Then take
F=z n+1 Q (the constant factor being irrelevant for this part of the
question) ; the equation becomes
( l -^^- 2z fz +n{n+1) y =0 -
For the second part, use § 143, viz.
F (a, /3, 7, x) I v?- 1 (1 - v)y-^~ l dv=-\ v*' 1 (1 - y)?-"- 1 (1 - xv)~ a dv,
Jo .'
and the result in Ex. 6, Misc. Exx., ch. vi (p. 81, supra) ; writing
1
2z = x + - ,
x
, u — 1
and v =
u + 1'
where w = cosh 6, we obtain the required expression for Q n n (x) as the corresponding solution
of the equation.
Let Un = - \ ,
so that, from the equation satisfied by „,
Again, differentiating the equation, we have
xy'" + {n + 2) y" + y = 0.
Let u __#5±lte).
9»+i W
also „' («) = — n+1 (a;), so that
"n+i — —
«'(#) '
and therefore n + 1 = #w m+1 .
Thus u n =
n + 1 - tfMjj+i
1 «
n+l-n+2-n+3
From the initial equation above, we have
- I u n dx
n (x) = Ae J"
To determine A, let x = ; then, as <£„ (0) = — , we have
nl
, . . I"/ X «nta + B sin (x + a) e - xcota .
2
The primitive of y" = ■ - % y
is y= A cot« + B(xcotx — 1).
iftc. 31. Let y = u (x 2 — iy n ; the equation for u is
(a? - 1) w" - (2?i - 2) aw' = (m + 1) (m + 2m) it.
Differentiate this equation n times, and write v = -r-^ ; then
(« 2 — 1) i/' + 2xv = (m + n)(m + n+ 1) a,
which, being Legendre's equation of order m + n, is satisfied by
fjm+n
Hence a value of u is
which gives the required value of K m (x).
To obtain the expression for L m (x), substitute in the w-equation
.= f—.dt,
J x — t
determining T if possible as a function of t alone, so that the
equation is satisfied. We must have
> ( ^^ + ^-2)/ ( -^ y2 ^ = /( W + l)( W + 2,)^^.
Integrating by parts both the integrals on the left-hand side until
only the first power of (x — £) _1 remains, we have
+
{ 2n-(m+l)(m+2n)}T-^{(2n+2)tT} + ^\(F-l)T}
dt = O r
the expression outside the integral sign being taken between
limits to be determined.
The subject of integration is
(*» - 1) T" - (2n - 2) tT' - (m + 1) (m + 2n) T,
which vanishes if T =u(t) = (t 2 - l) n K m (t) ;
DIFFERENTIAL EQUATIONS 213
and then the quantity outside the integral vanishes if the limits
are 1 and — 1. Consequently we have a solution
Ex. 32. (In the equation the sign of b s should be changed.)
Take y = ux m ; the equation for u is
„ 2m , ,„
u" H u'- bhb = 0.
x
Then (§ 136) we have (t) = - b 2 + t 2 , i|r (t) = 2m ; so a solution is
u = A\^ 1 (b 2 - t 2 )™- 1 dt
between constant limits determined by
[e xt (b 2 - t 2 ) m ] = :
that is, we take the limits to be b 2 and — b 2 . When t — b cos 6,
the limits of are to ir ; omitting a constant factor, we have
y = x m \ e tocos o sin 2 ™- 1 6 dd.
J o
Ex. 33. We have
y" + (m 2 +n 2 )y = I cos(ma;cos + nx sin ) 2 d.
Jo
Multiply by x, and integrate by parts : then
xy" + x(m 2 +n 2 )y= — (msin — ncos)sm(vixcos + 6)(m cos $> + n sin $) cZ$
J o
= — n sin (— ma + 0) — n sin (mx + 6) — -~,
and so xy" + y' + x (in 2 + n 2 ) y = — 2n sin 6 cos mx.
Ex. 34. Write « = 2z — 1 ; the equation is
z(l-z)y" + (l-2z)y'-±y = 0,
a hypergeometric series equation for which 7=1, a = ft = $.
Thus (§ 143) one solution is
A I v - (1 — v) 2 (1 — zv) 2 cfo.
J
Another integral is obtained by using any one of the equivalent
integrals in §§ 120, 121, 124, and noting that the foregoing integral
is a constant multiple of F (a, ft, 7, z) for the values of a, ft, 7.
The second part is differential substitution ; and the third is
algebraic substitution.
214 GENERAL EXAMPLES OF
Ex. 35. With the notation of § 136,
(t) = 2t, ^ (t) = t s + I ;
so a solution is given by
or, taking t = — 8' 2 , a solution is
within the limits [e " xff * e ~ ^ 6>] = 0.
One limit is = 0; the others are given by 6 6 = oc . Take 8 2 equal
to t 2 , cot-, ; take as the lower limit, and
the other three in turn as the upper limit.
Ex. 37. The indicial equation is p (p — p — 1 ) = 0. The relation
between successive coefficients is
_ ( P + fJ ,-m)( P + fi- n)
^-(p + H + Vip + H-p)^-
For p = 0, the series finishes when fi = n< p;
p = p + 1, the series finishes when p + l+fi = rn>p.
Ex. 3S. The primitive is
y = A (x + 1) + 5a 2 + Car.
Ex. 39. If the lowest powers of x in u and u are the same, say
ax m and cx m respectively, then writing
y = A'u + B' (u v) ,
we have new quantities it, u v, in which the lowest powers of *
DIFFERENTIAL EQUATIONS 215
are not the same. Accordingly, we can take (without loss of
generality)
u — x m (1 + positive powers of x), v = x n (1 + positive powers of x),
where m > n, and the coefficients of the lowest powers are absorbed
into A and B. Then as
u" + Pu + Qu = 0, v" + iV + Qv = 0,
, „ vu"—uv" « v'u" — u'v"
we have P = -, -, , (J = —
vu — UV V u — vu
AVhen the values- of u and v are substituted, and a non-vanishing
factor m — n is removed from the numerator and the denominator
of each fraction, we find
m + n — 1 4- positive powers of x
P =
Q = -
x(l + positive powers of x)
inn + positive powers of x
af(l + positive powers of x) '
Hence there is no factor « 2 in the denominator of P and no factor
x 3 in the denominator of Q. If m + n = 1, there is no factor x in
the denominator of P; and corresponding conditions will make
only x 1 or even x" in the denominator of Q.
Similarly for any factor x — a in the quantity uv — u'v : we
merely arrange u and v in ascending powers of x — a.
Ex. 40. One solution of the equation is /„ (ix) ; the condition
that this shall contain only integral powers of x is that a shall be
a whole number.
When a is a whole number, the other solution is given by
the results of § 105 ; it possesses a logarithmic infinity at the
origin.
Ex. 41. If the equation has a polynomial solution, arrange it
in descending powers of x, and substitute; denoting by x p the
highest power, a necessary condition is that p (a positive integer)
should satisfy the equation
Ap(p-1)+Dp+F = 0.
But the condition is not sufficient ; for it does not secure that the
descending series, which begins with x p , terminates with a positive
power of x.
216 GENERAL EXAMPLES OF
In order that the complete primitive should be a polynomial,
it is necessary and sufficient that
(i) both roots of the equation Ap (p — 1) + Dp + F=0 should
be positive integers ; and
(ii) E/B should be an integer not greater than unity, and
such also that 1 — (E/B) should be not greater than the greater
root of the preceding quadratic.
Ex. 42. The primitive is
v = Ax$ (1 - sc)i + Bxi (1 - %)i.
If v, and v 2 be two linearly independent primitives with constants
A, and B lt A 2 and B 2 , then
x [B 2 v,—B,v 2 y
1 — x \A 2 v,— A,v 2 ,
from which the result follows.
p. 568 Ex. 43. (i) Let ay? + 2hy,y 2 + by? = 1 : then
(ay, + hy 2 ) y,' + (hy, + by 2 ) y 2 ' = 0,
(ay, + hy 2 ) y," + (hy, + by 2 ) y 2 " + ay,'* + 2%/y/ + by," = 0,
so that, on substituting from the equation for y" and y 2 " and using
the earlier relations, we have
ay,'*+2hy 1 'y 2 '+by 2 '*=Q.
Hence £ -^ = (ay,' + hy 2 ') y," + (hy,' + by 2 ) y 2 "'
= -P(ay,'*+2hy,'y 2 '+by 2 '%
on substituting for y," and y 2 " ; that is,
<|| + 2PQ = 0.
Again, we have
where G is a non-arbitrary constant, so that
y± = v* =Cc -jPdx
%i + by 2 - (ay, + hy 2 )
Also (hy, + by 2 y + (ab - h?) y,*=b,
and therefore ^ T = Ce~ $ Pdx ,
{b-(ab-h?)y?}*
the integration of which gives y, ; and then
by 2 =-hy,+ {b- (ab - h>) y?f- .
DIFFERENTIAL EQUATIONS 217
(The method adopted for Ex. 44 can also be used for the pre-
ceding example by taking the relation in the form y x y 2 = a.)
(ii) We have
(2/i + b) y 2 + (y 2 + a) yl = 0,
and (y, + b) y 2 " + (y 2 + a) y," + 2 yi 'y 2 ' = 0.
In the latter, substitute for y" and y 2 " from the differential equation ;
then
2^1 'yi = Q (2t/, y 2 + ay x + by 2 )
= -Q(2c + ay, + by 2 ).
Differentiate again, and substitute for y" and y 2 ; then
Now y^-y iy :=Ge-l pdx ,
so that -^-y =^7^ r = s 5—
2/i + 6 - (2/2 + «) 27/!^ + aT/j + by 2
Ce-l pda;
- (2c + aj/i + %)
Accordingly
(2 C + a y + 62/^ (C " Ct6) = ~ yiV = ^ (2C + ayi + %2) '
and therefore
(2c + ay, + by 2 f = 2^ e -^ Pdx (c- ab),
which, with the initial equation, determines y Y and y 2 . When
their values and the values of y{ and y 2 are substituted in
(^ + 2P}y 1 'y 2 '=3Q(ay 1 ' + by 2 ') !
we have the required relation.
(iii) The values of y x and y 2 are given by the equations
,,J4.,,i_1 v y = JL e 2 JPda;
2/i + 2/2 _ - 1 ' 2/i2/2 2O 2 '
and the condition is
S'»*{(is- p «)' + i«1 + 6i> = °-
The analysis follows the foregoing lines.
218 GENERAL EXAMPLES OF
Ex. 44. By taking linear combinations of y u y 2 , y 3 , which of
course are integrals of the equation, the relation can be trans-
formed into yi = y,y 3 ,
so that we may take y 2 = ty 1 , y s = i 2 ^. We have
y"' + Py.' + Qyi = 0.
Since y 2 (= ty,) is a solution, we have
tyr + 3t' yi " + 3fV + f'% + P (ty,' + 1%) + Qty, = 0,
that is, St'yi' + Sf'y^ + ?//" 4- Pt'y 1 = 0.
Since y s (— Py^) is a solution, we have
t%'" + 6tt V + %/ ( 2tt " + 2t '*) + Vi i 2tt '" + 6 *'0
+ P(t%' + 2^1^) + Qt*y t = 0,
that is, Qtt'y" + %/ (tt" + i' 2 ) + y, (2tt'" + Qt't") + 2Ptt'y 1 = 0.
Eliminating P between the last two relations, we have
Gy.'t' 2 + et'f'y, = 0,
and therefore y± = —r ,
V
where A is any constant ; and therefore y 2 t' = At, y 3 t' = At 2 , so that
the equation is solved when a value of t is known.
Substituting this value of y t in the two foregoing relations,
we find /" t"n
r ~t' S t'*'
Hence f = 2Q;
and t is any solution of the equation {t, x] = \P.
The primitive is (B 1 + B 2 t + B s t 2 )/t'.
See a memoir by the author "Invariants,...," Phil. Trans.
(1888), pp. 377-489, §§ 81-84.
Ex. 45. Take s = y/z ; then
s ,^zy'-z'y
z' 2 '
s' zy' — z'y z
— ~i " ~ >
zs z
DIFFERENTIAL EQUATIONS 219
and therefore P = *- + SLzI\ a = - 2 - .
s z
Consequently g = _2^-f 2 gj = 2J + \P\
which is. the result.
Ex. 46. For first part, see § 58.
For second part, take (after § 110) y = ; the equation for z is
/' + Pz' + Qz = ;
and now use the first part of the question.
For the third part, we have Q — x — 1, P = — x ; so that
z" — z - x {z — z) = 0.
A particular solution of this equation is z = e x , so that a particular
integral of the Riccati equation is y = — 1 (as otherwise is obvious).
Hence (§ 110) we substitute
, 1
« = - 1 + -;
J v
and we find ve^ ~ 2x = A — je^ ~ 2x dx.
Ex. 47. When the variable is changed from x to z, the new
form of the equation is
y"-^(/n + 3|-3/ 1 «+/ 1 *y) + 3^(2/ 1 -i-/ 1 ',
-3y'(/iy-l)-y=0.
Choose / so that /j = - , and therefore /= log y. The equation now
becomes .„" _ y — o,
so that y = ^le i: + J Be- J!
= A & - +Bye~ x ;
y
Ae x
thus y = 1 _ Be _ x ■
Making the same change in the second equation and choosing
f— —y i > we obtain the transformed equation
d 2 y dy ,.
so that y = A+Blogz
= A' + B' logy.
220 GENERAL EXAMPLES OF
p. 569 Ex. 48. A first integral, under the given condition, is
and the primitive is
„ dr /, 8r 3 \ -
2 de^ r V-^
2r = c [sech {f (6 + a)}f.
Ex. 49. The two values of n are — ^-, — - 1 /- ; the primitive is
(a, b, c,f, g,h^x, y, 1) 2 =0.
(The equation is the differential equation of the general conic ; it
was first obtained by Monge.)
Ex. 50. (i) Write yx~ 2 =z, x=e e ; the equation is
so that f ~ Y = a + J z* + 2 I/O) dz.
d8)
(ii) The substitution for the first of the equations is
z = y(a + 2bx + cx 2 )~ 11 .
(iii) We have
b (ax 2 + 2hxy + by 2 + 2gx + 2fy + c)
= (by + hx +ff + Cx* + 2Gx + A,
so that, if Y= by + hx +f, we have
~=b*(Y*+Cx*+2Gx + A)-'i
and therefore
•5 d 2 Y 5 ( V
(Cx* + 2Gx + A)* ~=b§ \ r + 1
dx \(Cx* + 2Gx + A)i
which is of the preceding form.
Ex. 51. (i) y + & log (cos ^ + a)=B.
(ii) Write x — e 6 , y = ze^ e , p = dz/dd ; the equation is
*£-0 + »(?-«)-*
that is, ^(g-ig-^ + S-O.
DIFFERENTIAL EQUATIONS 221
This is satisfied by the two independent (but not simultaneous)
equations
S-if+'-^S-K— - *)•
which lead to the respective equations
a. .3. i
y 2 —Ax' 2 ' = x' i ,
y 1 — Bx 1 = —x 2 ;
but these equations do not coexist. Still, it would seem from
these forms, as if a primitive in finite form exists ; I have not been
able to obtain it.
(iii) A first integral is
p s — Spec + o> s — A' :
resolve the cubic for p, and use the method of § 18.
(iv) This is Eiccati's equation, § 108.
(v) A first integral is
9 + l
where x = e 9 , dyjdO = q ; resolve the quadratic for p, and use the
method of § 18.
(vii) A first integral is
where x = e e ; so the primitive is
I -r—« — r t = B+log x.
JAf + ^y+l *
A trivial solution is y = constant.
x + 2
(viii) The right-hand side should be n y 3 ; the primi-
\X -y- i- )
tive is
-e™ = B + Ae™-n ["L+l+A e ^dx.
y J x + 1
(ix) cosh- 1 ^ = B + A cosh"
x.
a
222 GENERAL EXAMPLES OF
p. 570 Ex. 52. The given curves are
(i + £)£«»«-K)*.»-«.
Their orthogonal trajectories are
1 + — r 2 -;- cos + r — - sin 6 = 0,
r 3 J dr v rv
that is, (r 2 + 2j)cos0 + (r-^)^sin0 = O.
Multiply by 2 sin 0, and integrate : then
r 2 + 2 - ) sin 2 = constant.
ifo. 53. The given curves are
dr
(r" — a™ cos n6) -j-
Their orthogonal trajectories are
(r n — a n cos nd) r 2 -
c
which can be expressed in the form
dr
(r" — a™ cos nd) -tq + a n r sin nd = 0.
(r n — a n cos «#) r 2 ^ a*V sin ?i# = 0.
dr
dd
n , n nr 11 " 1 cos nd — nr n sin nd -=-
n « cos «0 at* or _
r sin«0 air ?- n cos nt> — a n
Integrating, we have
r"sin??#
—p. = constant = cot da
dt~dt' ~dt~ dt + da ~di '
when these are substituted in the condition
dx d!; dy di) _
didi + ~di~di~ '
the general result follows.
The equation, for the particular example, becomes
da df .
a 2 — a^r-~- sin t = 0,
dt da
A , ^ . da df dt
that is,
a da sin t
hence the orthogonal system is
\ — —-da — log tan ht = constant.
J a da 5 2
Ex. 55. Use the property of Ex. 57, and take a, v as new
coordinates. The given system is
3<£ d dv _
du dv du
For an oblique trajectory
dv \du)
(S +tana
du , (dv \ ,
1 — -j- tan a
\duj
thus the equation of the oblique trajectory is
d(j) dip dv + du tan a _
du dv du — dv tan a
Ex. 5(3. We have x + ¥- + -+2v = 0,
x x
and so the differential equation of the circles is
x* x i x
224 GENERAL EXAMPLES OF
Thus the equation of the oblique trajectory is
(a?-f- c) + 2xy^— ^= 0,
where a is constant ; that is, the equation is
x 2 — y 2 — o + 2axy = p \a (x 2 — y 2 — c) — 2xy}.
In polar coordinates, the equation is
r ~„ {r 2 sin (l3-8)-c sin (/3 + 6)} = r 2 cos (/3 - 6>) - c cos (/3 + <9).
.#«;. 57. We have
du .dv ... . . 3m . dv . „ , . .
, 3m . 9w . /9w . dv\
hence r~ + i — =«h- + ir- ,
3?/ cty \ox ox)
so that
3m _ dv du dv
dy dx' dx dy'
, , , „ 3m dv du dv .
and therefore ;— - ^- + =- — = 0,
dx ox dy dy
which establishes the proposition.
In case (i), u + iv = log r + id, so that
u = log r.
In case (ii), u + iv = e x (cos y + i sin y), so that
m = e? cos y.
In case (iii), u + iv = cos -1 (a; 4- iy), so that
cos u cosh v = a;, sin u sinh v = y,
and therefore — r A—- = 1.
cos- m sin'tt
In case (iv), u + iv = tan -1 (« + iw), so that
x + y tan m tanh « = tan u,
y — x tan u tanh v = tanh v,
and therefore a; 2 + y 2 — x (tan m — cot u) — 1.
p. 571 -##. 58. We have 1 - a + a 2 = 1. Now
y 2 =(a+l)(x+l) + 2 {a(l+x + x 2 )}i,
2x +1
2wm' = a + 1 + V«
(1 + a; + a 2 '
DIFFERENTIAL EQUATIONS 225
hence
X + X 2
3
xy % a?y dy . . . x . x
Again, 2yy" + 2y' 2 = *Ja.
(1+x + x*)
a •
and 2/~ = ii;rx^ +
4 [a + * 1 + ax (i + cc + x ?f.
, [ g(l + a) 2Va
so that
2yy"-y'* = Ja 7 *— - s - f V« — - — -r - I j-£— ^
Hence vi/'-li/*-£ ^ ^ + a .^ =Q
ttence yy *y ? 1 -^ cte + 8 1 -a 3
Similarly for the other value of y.
Taking y = w 2 , we have as the equation for u
of which the primitive is
u = A {(a + a;)* + (1 + ob)*}* + -B {(a + a)* - (1 + ««)*}* J
consequently the primitive of the given equation is
y = [4 {(a + «)*+ (1 + «a;)*J*+5 {(a + «)*- (1 + aa)*}*] 2 .
Ex. 59. y = .Ae* + e** {5 cos ($« V23) + C sin (W 23 )} I
z = AeF + d x {B'cos^x V23) + C sin (^23)}/ '
where 24# = - VoB - 3 Vl3(7, 24(7 = 3 V23.B - 450.
ifo. 60. Primitive is
y = (Acos(j)+B sin <£) e _/
^ = (— A sin <£ + B cos 0) e _/
Ex. 61. Primitive is
« = * + 2 - %A'e~ l - (C + 2D't + SE'f) e*
y = 2 + A'e-t + (B' + G't + D't 2 + E't 3 ) e
F. IS
226 GENERAL EXAMPLES OF
Ex. 62. Primitive is
2 log (as + y + z) + = A, q = B, so that
y = Ax + A 2 - aB, z = Bx + AB ;
one solution, containing two arbitrary constants.
DIFFERENTIAL EQUATIONS 227
From (ii), substitute for q in the first equation ; then
y = x- + ixp + 3p*,
of which the primitive is y = f A 2 - \xA — \x*, leading to
az = -\A(x + A)*\
a second solution containing one arbitrary constant, and distinct
from the first solution.
The equation in y and a; has a singular solution 3y + x 2 = 0,
leading to az + ¥ y a? = ; a third solution containing no arbitrary
constant, and distinct from the other two solutions.
(For the relation between the solutions, see my Theory of
Differential Equations, vol. iii, §§ 197-201.)
Ex. 66. The primitive is
* = 2 A r e a r, y = 2 B r e f \ z= 1 C r e a -;
r=l r=l r = l
where X 1; X 2 , Xj are the roots of
a — X', h, g = 0,
h, b-X', f |
9> f c-X' [
and (a - X r ) A ,. + hB r + gC r = 0, hA r + (b - X r ) B r + fC r = 0.
When a -9h = h _hf =e _f^
f 9 h
the equation for X' becomes
( x-xy + ( x-xy(^ + ^ + h J) = o,
that is, the roots are, X repeated, and p. For the root X,
(a-X)A + hB + gC=0, hA + (b-X)B +/O=0,
both of which lead to the single equation
A B G A
f 9 h
For the root u,, we have a — a = - — — -j- , b- u=-!~— ^f = 0;
9 h j h
thus
A s (- ¥-&) + B 3 h + G 3 g = 0, A 3 h + B s (- 9 j - f {) + G 3 f= 0,
. B j, B n B
so that A 3 = -p , n 3 = — , 3 = -7- ,
leading to the primitive required,
15—2
228 GENERAL EXAMPLES OF
Ex. 67. I do not understand the result, as an answer to the
question.
Ex. 68. The equations of the family are
(x — z tan a) 2 + Xy 2 = u, z = v,
where u and v are parameters, and X is a fixed constant. Hence
along the curves, we have
dz = 0, (x — z tan a) dx + Xydy = 0,
so that (§ 160) P=Xy, Q = z tan a - x, B = 0.
The condition that they can he cut orthogonally, being the " con-
dition of integrability," is Xy tan a = 0, that is, a = 0, or the line of
centres is perpendicular to the planes of the conies.
Ex. 69. Use the method of § 164. The general primitive is
z-
+ x — y = A ;
x + y y '
and A = \ for the particular conditions.
Ex. 70. (i) (x + yY(y + z) = A;
(ii) J^- = A;
..... x + 2a?z .
(m) = A;
yz
p. 573 (iv) (xy - z 2 ) y 2 = A ;
(v) The. condition of integrability is not satisfied for the
given equation. It is satisfied for
(2x + yz) ydx — x 2 dy + (x + 2z) y 2 dz = 0,
of which the primitive is
— + xz + z 2 = A ;
y
. .. z x y .
(vi) - + -+•- = 4;
x x y z
(vii) x 2 (yz + xz — \z 2 ) = A;
, .... /sin x sin y\ . .
(vm) z H ^ )+smz = A;
V x y )
,. . xy — z 2 .
DIFFERENTIAL EQUATIONS 229
(x) x yz 2 + 2\og^-^ = A;
CD
, ., a? + y 2 + z (x - y)
(xi) * ^ U = J^ •
xy
, .., #2-1 .
(xn) « = .4 ;
v ' J yz-\ '
(xm) 2- - 2a; = A •
z
(xiv) a?y — xy 2 + a? — y s + z(x + y) = A;
yz — x 2
yz + y 2 + z 2
, , yz — x 2 .
(xv) — ^ = J..
ifo. 71. Take a, b, c = -r , -~ , ^ ; the equation becomes
A (B - C)xdydz + B(C- A) ydzdx + 0(A - B) zdxdy = 0,
which is the equation of the lines of curvature on
Ax 2 + By 2 +Cz 2 =l;
see § 168, Ex. 3, in the text.
Ex. 72. (i) A consecutive line through the point is given by
xu + yv + zw = 0, xdu + ydv + zdw = 0,
so that we may take
x = vdw — wdv, y = wdu — udw, z = udv — vdu.
A consecutive point on the line is given by
xu + yv + zw = 0, udx + vdy + wdz = 0,
so that we may take
u = ydz— zdy, v = zdx — xdz, w = xdy — ydx.
(ir) From the equation Ax l ~ n + By 1 - 71 + Cz 1 -* 1 = 0, we have
~ n dx+-dy + -dz = 0,
and therefore
A *l 9.
x n y n z n
ydz — zdy zdx — xdz xdy — ydx'
Thus the differential equation will be
ax n (ydz — zdy) + by n (zdx — xdz) + cz n (xdy — ydx) = 0,
if A a + Bb + Cc = 0. Hence the result.
230 GENERAL EXAMPLES OF
(iii) The transformed equation is
au n (vdw - wdv) + bv n (wdu — udw) + cw n (udv — vdu) = 0,
the primitive of which is
A'u 1 ~ n + B'v 1 -" + C'w l ~ n = 0, A' a + B'b + C'c = 0.
Eliminate u, v, w between the former equation and
axu n + byv n + czw n = 0, xu + yv + zw = 0.
Ex. 73. (i) z = (Ax + 4m 2 A m B m = (2m - l) 2 :
singular integral is z = 0.
(iii) z% = 3 {x (a + 1)}* + 3 {y (a - 1)}* + b :
no singular integral.
(v) {3 (1 + a) 2 + (1 - a) 2 zf- = 72 (1 - a) 2 O + ay + b).
. .. x + afy — toz _ fa; + (»w — a>V) .
d 574 ( V1 ) "7 E ^ — r _ !=-p -it ^^ r~ r. where a> is an lma-
p- 0< * v y (x + y—z)" 2 \(x + y-z) a )
ginary cube root of unity.
X
(vii) z = e*^ F(x + y).
( viii) 4»* = x (a 2 + a 2 )* + y (if - a 2 )* + a 2 log x + ( x2 + a ^ + h _
y+(y._ .)t
(ix) (*-2/ + ^) 2 = (* + 2/-i!')^(a:— 3y + z).
(x) 5 2 + ^ = J F(2/ 2 + ^ 2 ).
(xi) 2 = J. + x cos a + y sin a + \ log cosh 2y.
(xii) z 2 - \a (x - yf + \ (a + z) (x. + y) 2 =F(x + y + z).
(xiii) Use the transformation of §20 2. The equation becomes
XY(P -Q) + PX- QY-(X- Y) (PX + QY- Z\
of which the general integral is
(X,Y,Z) = -Z + (X+Y-l)f{(X + Y-lYXY} = Q;
then use the relations in § 202.
Or : — An integral of the Charpit equations is
px + qy — z = a (p + q - 1) ;
then proceed as in § 207.
(xiv) zx =f(y* + z%
DIFFERENTIAL EQUATIONS 231
(xv) An integral of Charpit's equations is p = ^ : the
primitive is
^- + i(ay-l) 2 +3^ + - 3 log(l-ay)=A
y + a 2a 3 a? a? ° v a '
z — x~-
1-ay
(xvi) z = scyfia? + xy).
(xvii) An integral of Charpit's equations is p = aq: the
primitive is
log z + \u (w 2 - 1)^ + \ log {u + (w 2 - 1)^} = B,
where u = (x + ay)/z.
(xviii) z" = (x + a) 2 + (y + af + B.
(xix) xz — F (3x — y 2 + y).
(xx) An integral of Charpit's equations is p 2 — q i = a?: the
primitive is obtained by effecting the quadratures in
dz = xds [ ~ ydy {a 2 (x> - y 2 ) + a«}* - J a™d log X ^ .
x — y- x + y
fax + &y + y^Y* jpjfaos + ^y+^zM
(xxi) j = H < j > ,
■ fax + fi 3 y + y s z) K * [fax + 3 y + y s z)^)
where the three sets of constants a, /3, y, X satisfy
Xa = aa + fih + yg,
X/3 = ah + /3& + yf,
\y = ag + fff + yc,
and the ratios a : /3 : 7 in each set are obtained as in Ex. 66.
(xxii) log (z — px — ay) —/log x = F {log (y — fix) — b log x),
a e ... ,, ae
where ^= 7 : P = r — >, «r(l-/)=c-
l-6 ; r-b-f "^ ■>' b-f
(xxiii) Use the transformation of § 202. The equation
becomes (X-Y)P+QZ=0,
of which the general integral is
4>(X,Y,Z) = (X-Y-Z)e~*-f(Z) = 0:
re- transform by the relations on p. 416.
232 GENERAL EXAMPLES OF
Or : — An integral of Charpit's equation is
px + qy — z = a :
then proceed as in § 207, obtaining the complete integral
z-x + a , , x + y
= o + a log .
x + y x
(xxiv) z{(Ax + y) 2 +A z +l}=B.
(xxv) When multiplied out, the equation is linear of
Lagrange's form, for which the subsidiary equations are
dx _ dy _ dz
z 1 + yz — xy y 2 + xy — xz x 2 — y 2 — xy — yz — zx'
Of these, two integrals are
X = \x 2 + yz + \y 2 = a, Y = \x 2 + xy + \z 2 =~b;
hence the primitive is F (X, Y) = 0,
where F denotes an arbitrary function.
[The differential equation, as given, can be expressed in the form
dXdY = dXdY
dx dy dy dx '
where -=- denotes complete differentiation with respect to x, when
z is regarded as a function of x and y ; and so for -j- . The result
follows.]
(xxvi) (a + ty^-j— 2 + -J- + .B.
(xxvii) z = xy + x cos 3 a + y sin 2 a + B.
(xxviii) \z +f(A)Y = \2xy + x 2 f(A)} B.
(xxix) x n +y n + z n =F (xyz).
(xxx) /0 2 + 1 + Az)i dz = x 2 + Ay 2 + B.
(xxxi) An integral of Charpit's equations is p 2 + q 2 = a 2 : the
general integral is
fxdx + ydy ,, „ ,i v „
* = J tf + y* K* 2 + 2/ 2 ) « 2 - a 4 "} 2 + c 2 ' 1 tan-' | + &
(xxxii) yz + x 2 =/(xz— ^y 2 ).
(xxxiii) 1 + 4>a 2 z =( h Ae* + B) .
DIFFERENTIAL EQUATIONS 238
(xxxiv) z = %Ax* + \B]f- + G, where AB + A + B = 0.
(xxxv) Substitute u = e x+ v, v = xy: the equation becomes
(if + ZW) — + (V + W2) =- = 1 — 2 2 ,
of which the general integral is
v — uz=f(vz — u).
Ex. 74. Writing <£> = z —px — qy —a, we verify that the relation p. 575
of § 208 is satisfied identically : hence the result.
Ex. 75. (i) The general primitive is
xz z* „ (z
obtained as for a linear equation.
For the first integral, take F ( - ) = — a b.
\yj y
yi y
For the second integral, take F (-)= — 2 (- ] ,so that it is an
instance of the general integral.
(ii) For the first, we have (z — a) p = 2, (z — a) q = 2b ;
elimination of a and b leads to the equation.
For the second, we have cyp = c 2 , cyg 1 (2 — a) = 2y; elimination
of a and c leads to the equation.
Taking the second equation in the form cy (z — a') = 2 + (z - ax) ~ = 0,
da
*(^!) + (a, b, c,f, g, h) = n'tf> (r, s, 1, a, - p, sp - ra)
= w«©(r, s, p, a);
,9© dd> dd> ,9© d
so n l ^r— = n J- — na -£ , n t ^ = n~-nr~ r ,
or da dk da- 9/ dli
,9© 9(6 9
n t ^— = n-~r — npr~. iv — = — ji^ + ns^f.
9s 96 r 9A 9p d<7 9/i
Substituting in the ©-expression for the necessary and sufficient
condition, and using the foregoing relation, we obtain the relation
in the required form
9d) d d(f>d(f> d_„
9a df 96 dg 9c dh
The second form of the equation is at once derived by using
transformation of variables.
The complete primitive of the last form of the equation, which
should be d ±t± d± _
da 9/ + dg ~ '
is b = Aa + Bf-ABg+C.
DIFFERENTIAL EQUATIONS 239
The general integral is given by this equation', combined with
C=(A, B)\
0-/-A.+ &
/
shewing that the general integral includes the complex of lines.
Ex. 86. With the notation of the example, the general expression
for the perpendicular being
1 _ , (du\ 2 1 fdu
f~ U ' + \W + iinMa0
the differential equation of the surface is
ft \ . ^"Vj. y ( du Y
/(»)-«-=y + sI ^y •
So, writing {/(") — ur\~ - da = dt,
we have (g)' + ^ (gj = 1.
7)t
Use 5 200 : we have =-r = sin a,
o ; the equation
becomes ^(iZ + P l )=P 1 (P- i - P 3 ).
Two integrals of the subsidiary system are
P 2 = AP X , P 3 = BP U
and these satisfy the condition for coexistence. Resolving for
Pi, -P2, P 3 , and writing
t = X 1 + AX 2 + BX s>
we have (A - B) ~ = \a> + {I a* + 4>a>Z (A - B))*,
whence the primitive follows at once.
240 GENERAL EXAMPLES OF
where u = (x 2 - a^) 2 (2^ + 2&' 2 + 2x 3 + 3z).
Ex. 88. Integrals of the subsidiary equations in § 221 are
p 2 = ap u p 3 =bp 2 ,
leading to (z - c) (1 + a 2 + b 2 ) = 2 (x 1 + ax, + bx 3 f,
which is the second solution.
If the first solution can be derived from it, the values of the
derivatives must be equal ; hence
2x 1
, x 1 + ax 2 + bx 3 _ _ z + 0,!
l + a 2 + b 2 ~ Pl ~-£ x*
+ a-i) 2
with two others. These give
Xl Xn
• = o", = era, — = ab
Z+Oi
so that
= u . — ULt t
z + «! z + a 2 z + a 3
Xj + ax 2 t bx 3 a
1 + a 2 + b 2 a 2 (1 + a 2 + b 2 ) '
and therefore a (x x + ax 2 + bx s ) = 1,
which on substitution for a, it a, ab gives the result. The proper
value of c is obtained by equating the values of z.
If the third solution is derivable from the second, the three
derivatives must be the same. This requires that values of a and
b will satisfy the three equations
2 V^^ 3 = 4*i + 2t V2 XlX3 1 .
! + « 2 + & (*,' + <**>*'
2a x, + ax 2 + bx 3 = ^ + 2 . x 2 x 3
1+a ' + 6 " {x 2 + x 2 f
2& i+ a '+6» = - 2* 3 + 2i V2 (^ 2 + 4- x 2 2 ) 2
X, '
x 1 [2 + i\/2x s (x, 2 + xi) -]
satisfy all three equations.
DIFFERENTIAL EQUATIONS 241
Ex. 89. The Jacobian condition for, coexistence is satisfied.
When the equations are resolved, we have
df d_i df_
dx dy dz
3 (ij --t- =F(t 2 + z") + a tan -1 - + b.
z
Ex. 91. We have (F lt F 2 )=p 1 p 2 p i ~2h = F* = 0,
(F u F t ) = 0, (F 2 ,F s ) = 0.
Resolving, we have F i = —p 1 tE 1 +p ll x s +p i % 4 ~+p i a: i ='0,a,nd. the two
distinct sets
p x p 2 - 1 = 0) p,p 2 +1=0]
p 3 -^4=0j' .p 3 +Pi=0J
A complete integral for the first set, is
2z = [x 1 (x 2 + ax 3 + a« 4 )p + b ;
and a complete integral for the second set is
2z = {« 2 (*i +~ax s — ax 4 )}^ + b.
In addition, there are the associated general integrals.
f. 16
242 GENERAL EXAMPLES OF
Ex. 92. We have (F u F 2 ) - identically.
The integrals of the subsidiary system for
2 (x 3 + x 4 )p 2 + x 2 (p 3 +p 4 ) =
are u = x lt v = x£ — (x 3 + # 4 ) s , w = x 3 — x 4 ;
and every integral of the equation is a function of u, v, w. When
u, v, w are taken as independent variables for the other equation,
it becomes
ou ov dw
The integrals of the subsidiary system are
v?v, wju ;
hence all integrals of both equations are functions of these two
quantities.
'*■ . 1 d'
Ex. 93. Let il denote a (-^ +- ~j : then
d ^ = e a (x*e k * 2 ),
du
da
\dx 2 x dxj
= e a {(2k + 4>kV) e kx ' ! + 2ke k * 2 }
= 4,ku + 4k 2 -^.
The most general integral of this equation is
uk = F (t - 4a J .
To determine F, let a = ; then
u = e** 2 ,
so that FQ = ke k * 2 ,
1 x -
or F(t) = - '
t
Consequently F I t — 4a j = —
■4y'*)i\ {3x - (x' 2 - iy'rf} 3 .
Similarly 17 = {as - (x' 2 - 4,y'*f} [Sx' + (V 2 - 4>y'*)*} s .
Thus we have the primitive of the /-equation.
244 GENERAL EXAMPLE'S OF
Let * = F(g) + Q\rf); then (§ 202) the primitive of the original
equation is given by eliminating x' and y between
3* 3 ,3* ,3 .
X = W> y = W z = x Tx' + y W~~
(v) z = \xy {(log xf - (log yY\ + xyF(^ + G (xy).
(vi) ]ogz = F(Xx-y)+G{\x + y).
{ vii) * + y + £ (a; - 2/) 3 f (*) = (p + q)] - (pz + yf f (p + q)
-2(qz + oo)(pz + y)[q'(p + q)-(l+pq)(p + q )]=0.
By the use of the former equation, this leads to
(x + y) (p' + qy') -(p + q) (/* + 2y') = 0,
on using z = p + qy'.
Now from the last, p' + q'y' = z" — qy";
and from the earlier equation, which was
zz' + xy + y = 0,
we have zz" + z' 2 + xy" + 2y' = ;
so that (x + y) (z" - qy") + (p + q) (zz" + xy") = 0.
Thus z" (x+y + pz .+ qz) + (px- qy) y" = 0.
But p z + v = (Z) + ir(Z),
with the foregoing value of Z.
16—3
246
GENERAL EXAMPLES OF
p. 579 Ex. 95. From (x — y) 2 *-^- + kz = 0, we have
J dxdy
Xo d 3 z . d 2 z dz f.
( x -y ) ^- 2(x -y ) dxty +K dy =0 -
Thea(i) (.- y y_^^- + -) + i c(_ + ^.0 ;
.... , No 3 2 / dz dz\ ( dz dz\ .
(11) (*-y*^\*di + v%) +K {"di + ydir >
Thus #"=- + 2/™ 3- is a solution of the equation for n — 0, 1, 2 ; it
is not a solution for other values of n.
Ex. 96. Take #'= \x 2 , y —\y 2 \ the equation becomes
d 2 z
dxty' ~~*\,
of which (§ 262) the integral is
hx-'-t
z = t{e *f(h)\.
For the second part, we have
p = - y I sin (*2/ cos ) cos $d<£,
J
s = — sin (xy cos d!d> — aw cos (xy cos 0) cos 2 cj>d,
Jo ' Jo
and therefore
s + xyz = — I sin (a;?/ cos <£) cos {jtdcfr + xy l cos («?/ cos ) sin 2 d<£.
Jo Jo
Integrate the second integral by parts ,' its value is
— sin (xy cos <£) sin $
+ I sm(awcos<£)cos<£d.
< J
Hence
s + xyz = 0.
DIFFERENTIAL EQUATIONS
247
Ex. 97. We have d £= j" cos f'd, ~ ^ T cos* ctf" dcj>,
= — / /"d ; hence
J o
oy
3 2 w 9V
+ S) = -//sin^/"^
a* 2 dy-
sin ./'
7T r-ir
— I COS =
J
3?/
For the second part, we have
dv
hence
d 2 v . d 2 v
dx
dv [* [*
^- = cos fd(f> + x cos 2 f'dcj},
000 J Jo
= 2 [" cos 2 f'd + x ("cos 3 ^/" dcj>,
Jo ' Jo
= — x I cos f" d<\> ;
Jo
dx 2
d*v
by
; d 2 v f" f"
,i + a-„ = 2 cos 2 4>f dcj}-x\ cos <£ sin 2 cf>f"d^
dy J Jo
= 2 J" cos 2 f dcf>
Jo
+
"- r(cos 2 4>-sin 2 <£)/'<%
Jo
cos (£ sin $/"'
and therefore
* (S + 9?) ~ ^ = /o * Sin2 & *+ "Jo C ° S #*+
sin$./
= 0.
hence
i?#. 98. (i) For the first equation, see Ex. 3, § 278, in the text,
(ii) Substitute z = Ae hx+k v ; then h? = &, P = A, so that
h = 01 1) &>) » 2 )
Ar = 0|' 1|' ft> 2 ]' »}'
s
S
* = 4„ + XA^+v .
fe. 99. We have a ^- = p + (xp' + yq' + zr') — ; so that, if
we have
D=a — xp' — yq' — zr,
du p du _ q du _ r
dx = D' dy = D' di~D'
248 GENERAL EXAMPLES OF
Similarly |? = - p' - (xp" + yq" + zr") £- ,
and so for the other derivatives. Thus
(du\ 2 (du\ 2 (d
) + Q' + (s)-*
\dx) \dy
and S = ^ + £ {xp " + yq " + zr " } -
Now
d 2 F_dFdy, cPF/du\*
da? du dx 2 du 2 \dx) '
d 2 F
and so S ttt = 0.
dx 2
G Gdu
Again n = - ^" J
° D p dx
so taking F 1 (u) = j — du,
G dF 1
we have -~ = -^— .
D dx
But, by the preceding part,
2^ =
dx 2 '
and therefore (|i + ^ + g) g 1 = 0. Thus
d 2 v d 2 v d 2 v n
1 1 =0.
3a; 2 dy 2 dz 2
Ex. 100. Let T = r 2m U^^,
where U n - im is a homogeneous function of degree n — 2m ; using
the theorem
dU dU dU , _ ,„
we have
V 2 T = 2m (2w - 2m + 1) r 2 ™- 2 U n -^m + r 2m ^ 2 TJ n _ mi .
Now take V as the most general polynomial of degree n,
and consider
w = F-4,r 2 V 2 F+ 4 8 r 4 V 4 7-u4 3 r 6 V 6 F+ ...,
DIFFERENTIAL EQUATIONS 249
we have V 2 w = V 2 F- A 1 {2(2n- 1) V 2 F + r 2 V 4 F}
+ A^ {4 (2n - 3) r 2 V 4 F-+ r 4 V 6 F} - . . . ,
so, taking - A 1 . 2 (2ft - 1) + 1 = 0,
A„.4>(2n-S)+A 1 =0,
we have V 2 u, = 0.
Further, write V 2 V = , where <£ is of degree n — 2. Then, by
the preceding part (as V 2 it = 0),
V 2 7=V 2 [J. 1 r 2 ^ 2 F-^ 2 r 4 v 74 F+...],
and therefore = V 2 [A^ty - 4 2 r 4 v 72 ^ + ...],
that is, the Particular Integral of the equation
is u = A l r 2 -A 2 r i V 2 (f> + A s r s V 4 ^-....
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