Differential Equations/Homogeneous x and y

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Not to be confused with homogeneous equations, an equation homogeneous in x and y of degree n is an equation of the form

F(x,y,y')=0

Such that

aⁿF(x,y,y') = F(ax,ay,y').

Then the equation can take the form

Which is essentially another in the form

.

If we can solve this equation for y', then we can easily use the substitution method mentioned earlier to solve this equation. Suppose, however, that it is more easily solved for ,

So that

y = xf(y').

We can differentiate this to get

y'=f(y')+xf'(y')y

Then re-arranging things,

So that upon integrating,

We get

Thus, if we can eliminate y' between two simultaneous equations

y = xf(y')

and

,

then we can obtain the general solution..