210 EQUATIONS OP AN ORDER HIGHER 
Suppose tlie equation reduced to the form 
d n y s($ l ~ x y\ 
dx n [dx^J 1 h 
d n ~^y 
then, assuming ^ n -\ — z i we have 
dz 
. , dz 
whence ax = >7-7 , 
/(*) 
x= Im +c (10) ' 
If, after effecting the integration, we can express z in terms 
of x and c, suppose z = (j) (x, c) we have finally to integrate 
d n ~\f . . . 
^»-1 = $ \ x i c ) (ll)j 
which belongs to Case 1. 
But if, after effecting the integration in (10), we cannot 
algebraically express z in terms of x and c, we may proceed 
thus. 
From dx^ 1 = Z} We ^ iave 
d n ^v 
d^= Jzdx 
_ f zdz 
~iWY 
d n ~ s y I" 7 f zdz 
dx n ~ 3 ~J dx ]f{z) 
_ f dz f zdz 
= Jf¥)J/^)’ 
and finally, 
f dz f dz [ zdz 
(,2) ’ 
mm