Let \(P=\begin{pmatrix}i\alpha&0\\0&0\end{pmatrix}\), \(Q=\begin{pmatrix}0&\beta\\0&0\end{pmatrix}\), \(v(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}\), and \(c=\begin{pmatrix}x_0\\y_0\end{pmatrix}\).
For full credit, your solutions should involve matrix exponentiation.
Let \(A=\begin{pmatrix}0&\alpha\\\alpha&0\end{pmatrix}\), \(v(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}\), and \(v_0=\begin{pmatrix}x_0\\y_0\end{pmatrix}\). Solve the differential equation \(\frac{dv}{dt}=Av\) with initial conditions \(v(0)=v_0\).
HINT: \(e^{At}=\begin{pmatrix}\cosh(\alpha t)&\sinh(\alpha t)\\ \sinh(\alpha t)&\cosh(\alpha t)\end{pmatrix}\).