The Fundamental Theorem : Given a system of differential equations d x /dt = f ( x ,t) if the components of f have continuous derivatives everywhere then then for any fixed T > T0 and I0 as N goes to infinity our approximation tends to the unique solution of this system with f (T0)=I0 .
Its nice to see this, and in particular the $N$ dependency, in action. In this example we see a sequence of approximation to a solution of dy/dt=-y^2+5sin(t) converging to an actual solution as N increases.
A careful reading of this theorem reveals three things:
1.That such a system of equations always has a solution for any initial data. This is often referred to as an existence result.
2. That such a system has a unique solution. This is often called a uniqueness result.
3. That our method tends to the actual solution, in other words that we can "trust" the numbers and graphs produced by maple. In order to estimate how big $N$ really needs to be is one of the topics covered during a course on "numerical methods".