Linear Equations with Constant Coefficients

Purpose:

To study n^th order linear differential equations with constant coefficients.

Description:

In this lab we show students how to solve linear differential equations with constant coefficients via solving the characteristic equation. We consider 4 cases for the possible zeros of the characteristics equation, real roots, complex roots, repeated real roots, and repeated complex roots.

We ask the students to use a CAS to solve the characteristic equation for some 3^rd, 4^th, and 5^th order equations. We also have them derive the differential equation of minimal order which are solved by the given functions.

Finally we have students graph the four functions cos( x), C₁ e^x, -C₁ e^x, and C₁ e^x cos( x), for fixed values of C, , and . They use their graphs to describe the effect of and on solutions of the equation y'' + y' + (² + ²) y = 0. They are also asked about the relationship between the graphs of cos( x) and e^x and the graph of the solution.

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