In an earlier blog post, I showed 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. In particular, I considered the system , where are real constants and are real variables, for the case where and . Now I look at the case where .
I assume the reader is familiar with the technique called the method of undetermined coefficients. (See, for example, the 8th edition of Elementary Differential Equations by Rainville, Bedient, and Bedient published in 1996 by Prentice Hall).
Given that and , there are three possibilities: (1) and , (2) and , and (3) and . For the first possibility, the solution is simple: and (since ), where and are arbitrary constants.
We now consider the case where and . From we have (since ). From we have .
We now use the method of undetermined coefficients to solve for . The solution to the homogeneous equation is , where is an arbitrary constant. (We will use this later.)
Now note that is a solution to the differential equation .
The equation expressed using differential operators is . Applying on both sides of this yields , where the last equality is explained by the previous paragraph. The solution to is . Note that this is of the form . From which we got earlier, we get , leading us to conclude that is a non-arbitrary constant.
We find the value of by plugging our solution into the original equation. That is, from we get . This yields .
Thus, the solution to the system when and is and .
The case where and is similar and is left to the reader as an exercise.