Extension to systems of ODEs

Sample system: $$\begin{align} u' &= a u + bv\\ v' &= cu + dv \end{align}$$

The Forward Euler method: $$\begin{align} u^{n+1} &= u^n + \Delta t (a u^n + b v^n)\\ v^{n+1} &= u^n + \Delta t (cu^n + dv^n) \end{align}$$