Solution to Homogeneous Differential Equation

Theorem
Let:
 * $\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$

be a homogeneous differential equation‎.

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

Its solution is:
 * $\ds \ln x = \int \frac {\d z} {\map f {1, z} - z} + C$

where:
 * $\map f {x, y} = -\dfrac {\map M {x, y} } {\map N {x, y} }$

Proof
From the original equation‎, we see:
 * $\dfrac {\d y} {\d x} = \map f {x, y} = -\dfrac {\map M {x, y} } {\map N {x, y} }$

From Quotient of Homogeneous Functions‎ it follows that $\map f {x, y}$ is homogeneous of degree zero.

Thus:


 * $\map f {t x, t y} = t^0 \map f {x, y} = \map f {x, y}$

Set $t = \dfrac 1 x$ in this equation‎:

where $z = \dfrac y x$.

Then:

This is seen to be 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 $z$.