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, .