Let the vector valued function $\vec x(t) =\begin{bmatrix} x_1(t)\\x_2(t)\end{bmatrix}$ have polar coordinates $x_1(t)= r(t)\cos(\theta(t ))$ and $x_2(t)=r(t)\sin(\theta(t))$, satisfying the differential equation
$$\vec x''=-\begin{bmatrix}\cos\theta\\2\sin\theta\end{bmatrix}/r^2$$
Where the derivative is taken with respect to $t$. Suppose $\vec x(0)=\begin{bmatrix}1\\0\end{bmatrix},\vec x'(0)=\begin{bmatrix} 0\\1\end{bmatrix}$.
I don't think I've ever seen a differential equation like this and have no idea where to begin. It can't be solved with the matrix methods that I know because it does not have constant coefficients. After two applications of the chain rule for the derivative I see now way forward.