I want to solve this system numerically, but I am stuck as to how to proceed. Do I need to transform this into a set of four 1st order equations before discretizing? Which numerical method would be recommended here?
$ \frac{d^2x}{dt^2} = \omega^2 x + 2\omega\frac{dy}{dt} $
$ \frac{d^2y}{dt^2} = \omega^2 y - 2\omega\frac{dx}{dt} $
Thanks!
Hint: Let $z = x + iy$, where $z$ is a complex-valued function. Then
\begin{align} \frac{d^2z}{dt^2} &= \frac{d^2x}{dt^2} + i\frac{d^2y}{dt^2} \\ &= \omega^2(x + iy) + 2\omega\left(\frac{dy}{dt} - i\frac{dx}{dt}\right) \\ &= \omega^2(x + iy) - 2i\omega\left(i\frac{dy}{dt} + \frac{dx}{dt}\right) \\ &= \omega^2z - 2\omega i\frac{dz}{dt} \end{align}
Solution: Rearranging terms
$$ \frac{d^2z}{dt^2} + 2\omega i \frac{dz}{dt} - \omega^2 z = 0 $$
The characteristic polynomial is
$$ r^2 + 2\omega ir - \omega^2 = (r + i\omega)^2 = 0 $$
which gives a double root, and a general analytic solution of
$$ z(t) = (c_1 + c_2t)e^{-i\omega t} $$
Take the real and imaginary parts to get
\begin{align} x(t) &= (c_1+c_2t)\cos(\omega t) \\ y(t) &= -(c_1+c_2t)\sin(\omega t) \end{align}