I’m currently teaching an undergraduate course on ordinary differential equations using the 8th edition of *Elementary Differential Equations* by Rainville, Bedient, and Bedient (published in 1996 by Prentice Hall). I’ve always wanted to have a blog post containing some LaTeX (using the plug-in WP LaTeX), so in this blog post I’ll be showing my solution to one of the exercises in the book.

I will derive the solution to a first-order linear system of two (ordinary) differential equations with constant coefficients for the case where the characteristic equation of the (constant coefficient) square matrix has repeated roots.

Let the system , where are real constants and are real variables, be represented in matrix form as or simply as .

I will show that this system has the solution

if and .

I assume the reader is familiar with the solution to the case where the characteristic equation has distinct real roots.

From the characteristic equation of , we get

and which has the solutions and

Now, if . That is, the characteristic equation has repeated roots if and .

The two roots yield two solutions and . (The complete solution would thus be .)

To find , recall that then use to get .

This yields two equivalent equations in and . (This can be seen by using in the second equation.) We get and letting and yields .

In finding , note that . Instead, assume that .

Substituting this into yields . Thus, and since for real .

Using yields the system

.

After a little handwaving, we get and . Thus, .

Finally,

.

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