## Numerically integrating a non-linear DE

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

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

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

For this integration method I found that $\displaystyle \alpha = 0.5$ yields the maximum local truncation error and that the multiplication factor $\displaystyle Q(h\lambda)=1+h\lambda+\alpha h^2 \lambda^2$. A stable integration is possible for$\displaystyle \alpha > \frac{1}{8}$ and $\displaystyle h < \frac{1}{\alpha}\cdot\frac{1}{|\lambda|}$. Now I'm given the non-linear DE
$\displaystyle \psi''+\tan^{-1}\psi=0$ with IC: $\displaystyle \psi(0)=0,\: \psi'(0)=1$. We can write this as a system of two first order DEs
$\displaystyle \begin{array}{rcl} x_1' =& x_2 &= f_1(t,x_1,x_2) \\ x_2' =& -\tan^{-1}x_1 &= f_2(t,x_1,x_2) \end{array}$
Now I have to give a boundary for the time step $\displaystyle h$ in which the method is stable in the neighbourhood of the ICs and to give the value of $\displaystyle \alpha$ for which a stable integration is impossible. I started finding the eigenvalues $\displaystyle \lambda_{1,2}=\pm i\sqrt{\tan^{-1}x_1}$, but I'm stuck there really.