# Using Inverse Problems as Projects in a Differential Equations Course

### Jeff Graham

Department of Mathematics and Computer Science
Western Carolina University
Cullowhee, NC 28723
USA
Phone: (704) 227-7245

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## ABSTRACT

Inverse problems are becoming increasing important in industry, but they do not receive much attention in the mathematics curriculum. To illustrate how inverse problems can be incorporated into a class, we present the following inverse problem as an illustration.

From data measured about x(t), determine m, b, k in the 2nd order ordinary differential equation

mx'' + bx' + kx = f(t)

subject to suitable initial conditions.

We present two approaches to this problem:

1. One can determine m, b, k via least squares from measuring \$x(t)\$ at various times.
2. One can calculate m, b, k as the solution to a nonlinear system of equations determined by measuring the amplitude of x(t) for various choices of the forcing function.

In both cases, we compare the results obtained using clean data to those results obtained using noisy data.

Keyword(s): differential equations