HOMOGENEOUS EQUATIONS, 
which is reducible to 
This is resolvable into two equations, viz, 
The first gives on integration 
Hence, since 
Hence, replacing 6 by log x, and z by 
c cx 2 2c ° y — (1 — c) x‘ 
the rational forms of the integral required. 
The factor u — 0 in (h) giving ^ = 0, or z = c, leads to the 
singular solution y = cx 2 . 
Class III. Equations which are homogeneous with respect 
dy d 2 y v 
t0 ^’ dx’ dx 2 ’ &C * 
Properly speaking, this class constitutes a limit to the class 
just considered. For when n becomes large, the quantities n, 
n — 1, n — 2, the supposed measures of the degrees of y, ^, 
approach a ratio of equality.