Solving $\dot{x} = -\pi\dfrac{A}{k}\sin(kx)\cos(\pi y),$ $\dot{y} = A\cos(kx)\sin(\pi y)$

Question

The following system is given: $$\dot{x} = -\pi\dfrac{A}{k}\sin(kx)\cos(\pi y)$$ $$\dot{y} = A\cos(kx)\sin(\pi y)$$ How can I find the parametric representation $x(t)$, $y(t)$?

Answer 1

You can first compute the ratio \begin{equation} \frac{\dot{x}}{\dot{y}} = -\dfrac{\pi}{k} \frac{\tan(kx)}{\tan(\pi y)} \end{equation} Which you rearrange as: \begin{equation} \frac{\dot{x}}{\tan(kx)} =-\dfrac{\pi}{k} \frac{\dot{y}}{\tan(\pi y)} \end{equation} Then you look for the solution of the system given by the two equations: \begin{equation} \frac{\dot{x}}{\tan(kx)} =K \end{equation} And \begin{equation} -\dfrac{\pi}{k} \frac{\dot{y}}{\tan(\pi y)}=K \end{equation} where K is a constant independent of x and y.The it is possible to solve the differential equations. For instance the first equation has solution \begin{equation} x(t)=\frac{1}{k} \arcsin(e^{k(K t +c_1)}) \end{equation} where $c_1$ is the integration constant.A siilar result is obtained for the second equation.