How to prove convergence in this methods of a differential equation?

Question

I have the differential equation $x'=A x$ with initial value $x_0$ different from $0$ where $A$ is the matrix $\pmatrix{0&1\\-1&0}$.

I have the following methods:

1. ) $X_{n+1}=X_n + hAX_n$
2. ) $X_{n+1}=X_n + {h \over 2}A(X_n+X_{n+1})$

The task is to prove that in 1.) $\|Xn\|\to\infty$ and in 2.) just analyze the behavior of $X_n$.

Answer 1

Note that $A^2=-I$, so that $A$ represents the complex unit $i$. Then compare 1.) to the evolution of $(1+ih)^n$ and 2.) to the evolution of $\left(\frac{2+ih}{2-ih}\right)^n$, and the exact solution to $e^{it}$. The qualities asked for are best seen in a polar representation.