Solution to Homogeneous Differential Equation

Theorem
Let:
 * $M \left({x, y}\right) + N \left({x, y}\right) \dfrac {\mathrm d y} {\mathrm d x} = 0$

be a homogeneous differential equation‎.

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

Its solution is:
 * $\displaystyle \ln x = \int \frac {\mathrm d z} {f \left({1, z}\right) - z} + C$

where:
 * $f \left({x, y}\right) = -\dfrac {M \left({x, y}\right)} {N \left({x, y}\right)}$

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

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 $t = \dfrac 1 x$ in this equation‎:

where $z = \dfrac y x$.

Then:

Clearly this is a differential equation with separable variables.

On performing the required integrations and simplifying as necessary, the final step is to substitute $\dfrac y x$ back for $z$.