Solving Initial Value Problems
with 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
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.
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