The Backward Euler scheme

We apply the Backward Euler scheme to each component equation: $$[D_t^- u = v]^{n+1},$$ $$[D_t^- v = -\omega u]^{n+1} \tp$$ Written out: $$\begin{align} u^{n+1} - \Delta t v^{n+1} = u^{n},\\ v^{n+1} + \Delta t \omega^2 u^{n+1} = v^{n} \tp \end{align}$$ This is a coupled $2\times 2$ system for the new values at $t=t_{n+1}$!