Question

How do double roots affect the particular equation for second order differential equations?

Answer 1

you may need to multiply by an extra t, or x, or whatever your independent variable is in order to keep the solutions linearly independent

for example in \(y''-2y'+y=e^t\) you will find that in using undetermined coefficients your particular solution will have to have a seemingly spurious \(t^2\) to work.

http://www.wolframalpha.com/input/?i=y%27%27-2y%27%2By%3De%5Et

This is to keep it linearly independent from the \(te^t\) term that will appear as a result of the double root