Solution to Homogeneous Differential Equation

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

be a homogeneous differential equation‎.

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

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

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


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

So, we set $$t = 1/x$$ in this equation:

$$ $$ $$

where $$z = y/x$$.

Then:

$$ $$ $$ $$ $$

... and we have converted it into differential equation with separable variables.

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