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HOMOGENEOUS EQUATIONS. 
Case III. When the given equation merely connects two 
differential coefficients whose orders differ by 2. 
Reducing the equation to the form 
d n y j.(d"‘y 
dx n f \jdx n ' 
Let then 
d*z .. . 
This form has been considered under Case I. 
It gives 
f dz 
x = I , ^ + C. 
(13). 
(2 ¡f(z)dz + GY 
If from this equation z can be determined as a function of 
x, G, and C',—suppose z = cf) (x, C, C 1 ),—then 
d^y 
dx n 
C, C), 
the integration of which by Case I. will lead to the required 
integral. If ^ cannot be thus determined, we must proceed as 
under the same circumstances in Case n. 
Ex. Given = 
ax ax 
Proceeding as above, the final integral will be found to be 
y = c x e a + c 2 e“ + c 3 x + c 4 . 
Homogeneous Equations. 
3. There exist certain classes of homogeneous equations 
which admit of having their order depressed by unity. 
Class I. Equations which, on supposing x and y to be each 
cly d 2 y 
of the degree 1, ~ of the degree 0, -~ 3 of the degree — 1, &c., 
become homogeneous in the ordinary sense.