Terminology for initial value problems.

A differential equation for a function that depends on only one variable, which is often time, is call an ordinary differential equation. The general solution to the differential equation includes many possibilities; the boundary or initial conditions are needed to specify which of those are desired. If all conditions are at one point then the problem is an initial value problem and can be integrated from that point on. If some of the conditions are available at one point and others at another point then the ordinary differential equations become two-point boundary value problems. Initial value problems as ordinary differential equations arise in control of lumped parameter models, transient models of stirred tank reactors, and generally in models where there are no spatial gradients in the unknowns.

Consider the ordinary differential equation

It turns out that we can prepare methods to solve sets of first-order equations, and that suffices to solve all initial value problems of whateer order. To see this, consider the high-order differential equation

with initial conditions

It can be converted into a set of first-order equations. Using

the higher-order equation can be written as a set of first-order equations.

The initial conditions would have to be specified for variables y1(0), ..., yn(0), or equivalently y(0), ..., y(n-1)(0). The set of equations is then written as

All the methods in this chapter are described for a single equation; however, when written in vector notation, the methods apply to multiple equations as well.

Take Home Message: All initial value problems can be solved using the method for a single, nonlinear initial value problem.