Solving Homogeneous Differential Equations

Homogeneous Differential Equations

A function f(x,y) is said to be homogeneous of degree n if we can express

f(x,y) = xⁿ .g(y/x)

A differential equation of first order and first degree is said to be homogeneous if it is of the form

dy/dx = f1(x,y)/f2(x,y)

where f1 and f2 are homogeneous functions of the same degree n.

Such an equation can be written as

dy/dx = [xⁿ .g1(y/x)]/[xⁿ .g2(y/x)]

=> dy/dx = F(y/x) ....(1)

Put y/x = v
=> y = vx

=> dy/dx = v +x.(dv/dx) .... (2)

From (1) and (2)

F(v) = v +x.(dv/dx)

=> F(v)-v = x.(dv/dx)

=> dx/x = dv/(F(v)-v)

=>dv/(F(v)-v) = dx/x

As this differntial equation is in separable form it can be solved