A differential equation of the form

*dy/dx = f(x,y)/φ(x,y)*

is called a homogeneous equation if each term of

*f(x,y)* and

*φ(x,y)* is of the same degree.

In such cases we put

*y = vx* and

* dy/dx = v + x (dv/dx)*
The reduced equation involves v and x only. This new differential equation can be solved by variables separable method.

The equations of the form

*dy/dx = (ax + by + c)/(Ax + By + C)*
can be reduced to the homogeneous form by substituting

x = X + h and y = Y + k (h,k being constants)

∴

*dy/dx = dY/dX*
The given differential equation reduces to

*dY/dX = [a(X + h) + b(Y + k) + c]/[A(X + h) + B(Y + k) + C] *
*= [aX + bY + ah + bk + c]/[AX + BY + Ah + Bk + C]*
Then the given equation becomes homogeneous,

*dY/dX = (aX + bY)/(AX + BY)*
Case of failure: if a/A = b/B then the values of h,k will not be finite.

a/A = b/B = 1/m

A = am, B = bm

The given equation becomes

*dy/dx = (ax + by + c)/[m(ax + by) + c]*
Now put ax + by = z and apply the method of variables separable.