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.

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