## Numerically integrating a non-linear DE

Given a DE $y'=f(t,y), \: t>0$ for a function $y(t)$ with IC $y(0)=y_0$. We have the numerical integration method

$w_{n+1}=w_n+hf(t_n+\alpha h,w_n + \alpha hf(t_n,w_n))$

Here $w_n$ is the numerical approximation of $y_n$ after $n$ equidistantial time steps $h$.

For this integration method I found that $\alpha = 0.5$ yields the maximum local truncation error and that the multiplication factor $Q(h\lambda)=1+h\lambda+\alpha h^2 \lambda^2$. A stable integration is possible for $\alpha > \frac{1}{8}$ and $h < \frac{1}{\alpha}\cdot\frac{1}{|\lambda|}$. Now I'm given the non-linear DE
$\psi''+\tan^{-1}\psi=0$ with IC: $\psi(0)=0,\: \psi'(0)=1$. We can write this as a system of two first order DEs
$\begin{array}{rcl}
x_1' =& x_2 &= f_1(t,x_1,x_2) \\
x_2' =& -\tan^{-1}x_1 &= f_2(t,x_1,x_2)
\end{array}$

Now I have to give a boundary for the time step $h$ in which the method is stable in the neighbourhood of the ICs and to give the value of $\alpha$ for which a stable integration is impossible. I started finding the eigenvalues $\lambda_{1,2}=\pm i\sqrt{\tan^{-1}x_1}$, but I'm stuck there really.