 ## Solving Initial Value Problemswith a Computer Solver

### A Quick Recap

Recall that when solving a differential equation alone we are typically led to a family of solutions, determined by the presence of one or more constants.

In order to narrow this infinity of solutions to a unique solution it is necessary to impose one or more initial conditions, depending on the order of the differential equation. When this is done we say that we are now working with a an initial value problem.

### The Graphical Approach

We have already seen how we can use a graphical approach to solving initial value problems. We can use the computer to create a slope field, then locate the initial condition within the slope field, and use the field marks as a guide in generating a curve which is a "rough" picture of the solution. Unfortunately this approach has a few drawbacks:

• The computer generated graph may not be entirely accurate, since there is an element of guesswork involved, and the grid used to generate the slope field's field marks can never be "infinitely refined."
• The solution we generate this way is just a picture. In many situations it would be much nicer to be able to have a function defined by a formula as the solution if at all possible.
• The graphical method works for first order equations only, but many important physical models are second order or even higher.

### Using A Solver

Thankfully there are many solver packages that have been created for use on personal computers that deal quite well with solving initial value problems. These solver packages fall into two groups: symbolic solvers, and numerical solvers. Some solvers can even fall into both groups, as we shall see in a later laboratory. In this laboratory, however, we shall concentrate on symbolic solvers.

So let's get down to business, and see how Mathematica may be used as a symbolic initial value problem solver. If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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