Differential equation in polar coordinates

Question

I have the following system:

$\frac{dx}{dt} = 3x + y - x(x^2+y^2)$

$\frac{dy}{dt} = -x +3y -y(x^2+y^2)$

Converting this to polar coordinates gives us:

$\frac{dr}{dt} = r(3-r^2)$

$\frac{d\theta}{dt} = -1$

This gives us a solution $\theta(t) = -t + \theta_0$. What would the solution for $r(t)$ be though?

Answer 1

The solution to $\dot{r} = r (3-r^2)$ is

$$ t = \int \frac{1}{\dot{r}} \,{\rm d}r + C = \int \frac{1}{r (3-r^2)} \,{\rm d}r + C $$

$$ t = \frac{1}{3} \ln\left(\frac{r}{r_0}\right) - \frac{1}{6} \ln\left(\frac{r^2-3}{r_0^2-3}\right) $$

with the initial condition $r(0)=r_0$

$$ r = \frac{\sqrt{3}\, r_0\, {\rm e}^{3 t} }{\sqrt{r_0 {\rm e}^{6 t}+3-r_0^2}} $$

Answer 2

You have $$\frac{1}{r (\sqrt{3}-r)(\sqrt{3}+r)} dr = dt.$$ Use partial fractions on the left hand side and integrate.