216 
HOMOGENEOUS EQUATIONS. 
To effect the proposed reduction assume x = e e , y = e n0 z. 
The transformed equation will he free from 0, and, on assum 
ing ~ = u, will degenerate into an equation of a degree 
lower by unity between u and z. 
It is easy to establish that, if the given differential equa 
tion be 
F ( x >y> dx’ dfy~° ( 23 )’ 
the reduced equation for determining u will be 
F{ 1, z, u + nz, u ~ + (2n— 1) u + n (n — 1) z) =0...(24) 
Suppose that by the solution of this we find 
U = (j) {z, c) 
(25), 
then since 
dz , 
u — ^ , we have 
dQ = 
6 = 
dz 
<f> ( z > c ) ’ 
dz 
0 (*> c ) 
+ C 
(26), 
V 
in which it only remains to substitute log x for 9, and for z. 
Ex. Given x 4 ~~ = {x 3 + 2xy) ^ — 4y 2 . 
This equation proves homogeneous on assuming x to be of 
the degree 1, y of the degree 2, ^ of the degree 1, and 
of the degree 0. 
Changing then, according to the formula (24), x into 1, y 
. dy . „ -* d?y . , du n . 
into z, into u + 2z, and into u + 3m + 2z, we have 
du 
dz 
+ 3u + 2z = (1 + 2z) {u + 2z) — 4s 2 
(«)»