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.

[tex] \displaystyle \frac{dy}{dx} = \frac{ y^3 + x^2y -x -y}{ x^3+ xy^2 -x+y}[/tex]

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

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