EQUATION - AND ITS TRANSFORMATIONS. 
799 
d~v 2p dv „ 
dx: 2 x dx ^ V 
when p is an integer, either positive or negative, is given by any one of the formulae 
„ ,, /1 d \i° +1 . , . 
,=x \r d,:) ( C 1 e ff-r +c 3 e ax ), 
1 d\-p 
x dx) 
(cp‘ u +Cr,e~ ar ), 
v=x ^i/h ±y 
\xdx) 
V — 
1 d \~P - 1 ( c x e, ax + ctf~ a 
x dx 
Putting now x=nz n , where n =2 p -f- 1 and q=~, the differential equation becomes 
7b 
d 2 v 
dz 2 
—a 3 z 3 ? z v — 0 
and the integrals take the forms 
( 4 ); 
/ /7\® + l c i z q 
v=z(z~ 2rj+] j) {c^i +c 2 e 9 ), 
d: 
d\~P 
V = (Z 3 ? +1 qr) ( c i e * 2 + c 2 e q ' q ) 
V = Z[Z 2 ? +1 
dz 
d\P 
di 
_ z q — - Z q 
+ c 2 e q 
Zi 
v — ( z -2 S +il\ p V c ^ Zq + c ° g ^ 
1 
If p is a positive integer =i, so that <2=.,- q, then, from the first and third forms, 
, g Al dV +1 / ?** , 
or 
v~z[z 2?+1 
iv 
(/■ 
29 -- _ 2? 
c 1 e 9 + c 3 e ® 
zi 
