I have the following ODE: $$\dot x=y+ax(x^2+y^2), \dot y=-x+ay(x^2+y^2)$$ and I want to prove that the system is equivalent with the following $$\dot r=ar^3, \dot \theta=-1$$
I start by taking polar coordinates but when I make the change I get $$ r \dot r=ax^4+2ax^2y^2+ay^4$$ Any suggestions?
If you differentiate the equations $r^2=x^2+y^2$ and $\theta=\tan^{-1}(y/x)$, with respect to $t$ you obtain the following equations: $$ \dot{r}=\frac{x\dot{x}+y\dot{y}}{r} $$ $$ \dot{\theta}=\frac{x\dot{y}-y\dot{x}}{r^2}. $$
If you make the substitutions on this equation with your ODE, you directly get the statement.