 Solution

For the test equation $y^{\prime }=\lambda y$, write one numerical step asThe linear stability domain isA method is A-stable when $\{z:\Re z\le 0\}\subseteq \mathcal{S}$.

Forward Euler has $R(z)=1+z$, soThis disk does not contain the whole left half-plane, so forward Euler is not A-stable. Backward Euler has $R(z)=(1-z{)}^{-1}$, soIt contains the left half-plane, and backward Euler is A-stable.