Suppose we have a system of ODE's: a${a}^{\prime}=-a-2b$ and ${b}^{\prime}=2a-b$ with initial conditions $a(0)=1$ and $b(0)=-1$.

How can we find the maximum value of the step size such that the norm a solution of the system goes to zero (if we apply the forward Euler formula)?

Edit: the main part is to calculate the eigenvalues of the following matrix, based on the Euler method, this becomes

$\left(\begin{array}{cc}-1-h& -2-2h\\ 2+2h& -1-h\end{array}\right)$

The eigenvalues are $(-1+2i)(1+h)$ and $(-1-2i)(1+h)$

How can we find the maximum value of the step size such that the norm a solution of the system goes to zero (if we apply the forward Euler formula)?

Edit: the main part is to calculate the eigenvalues of the following matrix, based on the Euler method, this becomes

$\left(\begin{array}{cc}-1-h& -2-2h\\ 2+2h& -1-h\end{array}\right)$

The eigenvalues are $(-1+2i)(1+h)$ and $(-1-2i)(1+h)$