Ode

3D-Filmstrip knows how to calculate and display solutions of the initial value problem for first and second order systems of ordinary differential equations (ODE) in one, two, or three dependent variables. Let us recall briefly what this means. We will be dealing with vector-valued functions x of a single real variable t called the “time”. Here x can take values in R, R2, or R3. The problem is to find x from a knowledge of how x0 depends on x and t (in the first order case) or a knowledge of how x00 depends on x, x0, and t (in the second order case). Thus in the first order case the ODE we are trying to solve has the form x0 = f(x, t) and in the second order case it is x00 = f(x, x0, t). In the first order case, the so-called Local Existence Theorem for First Order ODEs tells us that, provided the function f is continuously differentiable, given an “initial time” t0, and an “initial position” x0, then in some sufficiently small interval around t0, there will be a unique solution x(t) to the ODE with x(t0) = x0. There is a similar local existence theorem for second order ODEs (which in fact is an easy consequence of the first order theorem). It says that given an initial time t0, an initial position x0, and an initial velocity v0 then, in some