I was given the following second-order differential equation,
and that the solution is . Using the solution I determined that
Following from this I transformed this second-order differential equation into a system of first-order differential equations, which is
Now I want to perform a single step with starting from t=0 with the Forward Euler method and after that with the Backward Euler method. Firstly with the Forward Euler method I use:
and I compute as
Now I want to perform the Backward Euler method.
From which i get
y two results seems to be quite differnt and that gets me to believe that I have made a mistake somewhere. Could someone let me know if they believe this to be correct, or why this could be wrong?