Numerically integrating a non-linear DE
Given a DE for a function with IC . We have the numerical integration method
Here is the numerical approximation of after equidistantial time steps .
For this integration method I found that yields the maximum local truncation error and that the multiplication factor . A stable integration is possible for and . Now I'm given the non-linear DE
with IC: . We can write this as a system of two first order DEs
Now I have to give a boundary for the time step in which the method is stable in the neighbourhood of the ICs and to give the value of for which a stable integration is impossible. I started finding the eigenvalues , but I'm stuck there really.