## Mathematics & Science
Learning Center |
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## Solving "Impossible" Differential EquationsWe have learned theorems that guarantee to us that under the right conditions
an initial value problem has a solution that both exists, and is unique on some
interval containing the What this usually means is that the solution we seek cannot be written in a
form that uses a combination of the standard "elementary" mathematical functions
that we are used to. This may come as a shock at first, but absolutely the same
problem exists for the majority of functions you might sketch by randomly drawing
a functional graph on a piece of paper. In fact we might state this fact even
more strongly: So what then? Should we just banish these "impossible" differential equations
from our consideration and pretend that they don't exist? That would be fine,
but all too many of them are the very equations which describe "real life"
physical phenomena. These are the differential equations we have to solve to
model the heat flowing through a steel rod, or put a space shuttle into orbit. We
can't just The somewhat unsatisfying truth is that we must find approximate numerical
solutions to most actual applied differential equations. Now remember, these
solutions will not be given by the kind of formulas we are used to. Rather,
Some numerical methods play "join the dots" with these points to mimic, as
best they can, the behavior of the actual solution. One note about this kind of
solution: it can't contain any arbitrary constants. We get one solution, not a
family of solutions. This means that the problems we solve must be Later in the course we will learn how some of these numerical methods actually
work, but for now we'll let |
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ODE Laboratories: A Sabbatical Project by Christopher A. Barker©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA e-mail: cbarker@deltacollege.edu |