I am trying to perform a local stability analysis for the differential equation
$$dy/dt = tcos(y)+e^{-t}$$
I want to determine the step size I will need at a particular time t and corresponding values $y_s$ if I try to integrate this problem with a forward Euler method. I am also interested in what happens to the step size at large times.
The approximations, using forward Euler method, I am getting are $$ y_n = y_0+ h*\sum_{i=0}^n t_icos(y_i)+e^{-t_i} $$
where h is the step size.
Where I am currently stuck is finding a condition, dependent on h, where $y_n$ will be bounded as n approaches infinity. Furthermore, I do not know how to continue after I find this condition. Any help would be much appreciated.