Homogeneous Differential Equations

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.

Equations reducible to homogeneous form

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.


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