I'm trying to solve a system of coupled second order differential equations. I never did that before. I'm not sure where to begin.
The equations are:
$$\ddot{x} = -\omega\dot{y} - \frac{k}{m}\dot{x}$$ $$\ddot{y} = \omega\dot{x} - \frac{k}{m}\dot{y}$$
I'm wondering what method should I use to solve this and can I use the same method for all the coupled systems?
I know x should be: $$x(t)= -\tau V_{0x} \cos \beta e^{\frac{-t}{\tau}} \cos(\omega t + \beta) + x_0 + \tau \cos^2\beta v_{0x}$$
However, I didn't find how to process to get this solution.
I have those values at $t=0$: $$ v_x(t=0) = v_{0x} , v_y(t=0) = 0, x(t=0) = x_0, y(t=0) = y_0. $$
Finally, to lighten. $$\tau = \frac{m}{k}, \omega=\frac{qB}{m}, \tan(\beta) = \omega \tau$$
Any help would be really appreciated.