EXACT DIFFEEENTIAL EQUATIONS. 219 
as would directly result from (29) and (30). Expressed in 
the form 
du u _ f b\ 2 
dx a\/(d-\-a?) V a) U ’ 
this equation is seen to belong to the class discussed in Chap. 
II. Art. 11. 
On comparing the above classes of homogeneous equations 
we see that Class II. is the mosl general. It includes Class I. 
as a subordinate species, and Class ill. as a limit. 
It is proper to observe that Classes I. and n. are usually- 
treated by a different method from that above employed. 
Thus, in Class I., it is customary to make the assumptions 
y = xt, 
d 2 y _ v 
dx* x ’ 
d 3 y _ w 
dx 3 x 2 ’ 
&c. 
On substitution x divides out, and there remains an equation 
involving y and the new variables t, u, v, w, &c., which may be 
reduced by successive eliminations to a differential equation 
between two variables, and of an order lower by unity than the 
equation given. But this method is far more complicated than 
the one which we have preferred to employ. 
Exact Differential Equations. 
4. A differential equation of the form 
AI x d JL il il\- 
9 1 ’ y ’ dx ’ d-j? dx' 
0. 
(33), 
is said to be exact if, representing its first member by V, the 
expression Vdx is the exact differential of a function U, which 
is therefore necessarily of the form yjr (^x, y, — ... 
Thus ( ~- — yx 2 — xy 2 = 0, is an exact differential 
CLtXs CL0(s CbtX} 
equation, its first member multiplied by dx being the differen 
tial of the function ~ — cc 2 y 2 j-, and the first member it 
self the differential coefficient of that function.