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.