# Separating the variables

We should all be familiar with the method of separation of variables for first order ordinary differential equations. Here’s a neat example from John Starrett’s article in Amer. Math. Monthly.

$\displaystyle \frac{dy}{dx} = \frac{ y^3 + x^2y -x -y}{ x^3+ xy^2 -x+y}$

Using polar coordinates $x= r \cos \theta$ and $y=r \sin \theta$, the differential becomes
$dy = y_r dr + y_\theta d\theta = \sin \theta dr + r \cos \theta d\theta.$
Putting the other parts into the equation, we get
$\displaystyle \frac{ \sin \theta \frac{dr}{d\theta} + r\cos \theta}{ \cos \theta \frac{dr}{d\theta} – r\sin \theta} = \frac{\sin \theta (r^2-1) – \cos \theta}{ \cos \theta (r^2-1)+\sin \theta}.$
Rearrange and voila! We get the separable
$\displaystyle \frac{dr}{d\theta} = r(1-r^2).$

This entry was posted in Calculus/Analysis, Problems. Bookmark the permalink.