The Fundamental Theorem

This result will be at the very heart of this course. It tells us that the method we are using to approximate a solution to a system of differential equations on maple really works! Recall the method was simply a careful articulation of what a differential equation tells us, namely if we know what time it is and where we are then the equation tells us (approximately) where we will be after a small increment of time (the Deltat in the program). To see this method in action look at any of our maple templates In the statement of the theorem we use the terms mean T,T_0,I_0 and N to mean exactly what they meant in these programs and in lecture lecture, namely T_0 is the initial time, T is the final time, I0 is the initial position, and N is the number of increments that T-T_0 has been divided into.

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".