I was fooling around with and thought about how to use complex numbers in the case when is not a solution to the related homogenous equation. It then hit me: it is really quite simple.

First notes the following: and .

Then it is a routine exercise to see the following: given that are NOT solutions to is the characteristic equation of the differential equation. Then: attempt Put into the differential equation to see .

Then: if the forcing function is , a particular solution is where . If the forcing function is , a particular solution is where .

That isn’t profound, but it does lead to the charming exercise: if are NOT roots to the quadratic with real coefficients , then is real as is .

Let’s check this out: . Now look at the numerator and the denominator separately. The denominator: Now note that every term inside a parenthesis is real.

The numerator: is clearly real.

What about ? We need to only check the numerator: is indeed real.

Yeah, this is elementary but this might appear as an exercise for my next complex variables class.