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}<br /> x_1' =& x_2 &= f_1(t,x_1,x_2) \\<br /> x_2' =& -\tan^{-1}x_1 &= f_2(t,x_1,x_2)<br /> \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.