Express the system : $x'=y-x^3, y'=-x +y^3$ via polar coordinates

Question

I have trouble expressing the following system via polar coordinates :

\begin{cases} x'=y-x^3 \\y'= -x + y^3\end{cases}

so for :

$$\begin{cases} x=r\cosθ \\ y=r\sinθ \end{cases}$$

we get :

$$\begin{cases} (r\cosθ)' = r\sinθ - r^3\cos^3θ \\ (r\sinθ)' = -r\cosθ + r^3\sin^3θ \end{cases}$$

My problem here is, I don't know how to calculate the expressions $(r\cosθ)'$ and $(r\sinθ)'$, since the differentation was carried out with respect to $x$ and $y$.

I would really appreciate if anyone could help me out with finalising the expression to polar coordinates.

Answer 1

You have by product rule and chain rule: $$ \begin{split} x' &= \frac{dx}{dt} &= \frac{d[r\cos \theta]}{dt} &= \frac{dr}{dt} \cos \theta + r \frac{d[\cos \theta]}{dt} &= \frac{dr}{dt} \cos \theta - r \sin \theta \frac{d\theta}{dt} \end{split} $$ and you can use the same for $y'$.