Is that first-order approximation appropriate for iteratively solving the Diff EQ? Or does a higher order method such as Runge-Kutta have to be used? I notice the graphs don't align perfectly...
It's the easiest approach to solve the ODL: Euler Forward (yes, first order). The approximation could be much better and almost exact with a smaller step size. Runge-Kutta is used for higher ODL's, where the Euler method would fail (especially with periodic solutions).