Use matrix methods to find the general solution of the equations $$\dot x=3x+2y,\space \dot y=-5x-3y$$ Sketch the phase-plane trajectories in the vicinity of the origin.
I wrote the equations as $\mathbf {\dot X}=\begin{pmatrix}3 & 2\\-5 &-3\end{pmatrix}\mathbf X$ where $\mathbf X=\begin{pmatrix} x\\y \end{pmatrix}$. I eventually get $$\mathbf X= A\begin{pmatrix} -2\\3+i \end{pmatrix}e^{-it}+B\begin{pmatrix} -2\\3-i \end{pmatrix}e^{it}$$ As the eigenvalues here ($i, -i$) are purely imaginary with have ellipses centred at $(0,0)$. My question is:
How do we determine the handedness of the ellipses (and also if possible details of them such as major/minor axis, but I'm not sure this is needed)?
Thank you
