Consider the system of ODEs
$$y' = \begin{bmatrix}-6&4\\4&-6\end{bmatrix}y, \quad t\in[t_0, t_e], \quad y(t_0)=y_0$$
I'm asked for what stepsize the explicit Euler method generates a stable solution.
If it would be one ODE I would just find for what stepsize $h$ the stability function $R(z)$ in $$y_{n+1} = R(\lambda h) y_{n}$$ would be less than $1$, but I get confused when it's a system of ODEs. Do I write
$$y_{n+1} = \left(1 + h \begin{bmatrix}-6&4\\4&-6\end{bmatrix}\right) y_n$$
and use the eigenvalues somehow?
The eigenvalues of the matrix in the LDE are $λ=-2, -10$. The explicit Euler method is stable for real eigenvalues if $\lambda h\in(-2,0)$ for all $λ$. om this case this means $h<0.2$.