Solution to Homogeneous Differential Equation

Theorem
Let:
 * $\displaystyle M \left({x, y}\right) + N \left({x, y}\right) \frac {\mathrm dy} {\mathrm dx} = 0$

be a homogeneous differential equation‎.

It can be solved by making the substitution $z = \dfrac y x$.

Proof
From the original equation‎, we see:
 * $\displaystyle \frac {\mathrm dy} {\mathrm dx} = f \left({x, y}\right) = - \frac {M \left({x, y}\right)} {N \left({x, y}\right)}$

From Quotient of Homogeneous Functions‎ it follows that $f \left({x, y}\right)$ is homogeneous of degree zero. Thus:


 * $f \left({tx, ty}\right) = t^0 f \left({x, y}\right) = f \left({x, y}\right)$

Set $t = 1/x$ in this equation‎:

where $z = y/x$.

Then:

Clearly this is a differential equation with separable variables.

Once we have performed the integrations and done whatever tidying up is needed, we substitute $y/x$ back for $z$ and the job is done.