The differential equation system is
To solve this differential equation requires a particular solution xp(t) and two fundamental solutions of the homogeneous equation corresponding X1(t) and X2(t)
I find X1(t) and X2(t) for the following cases
r -> imaginary part
How do I find the particular solution xp(t) ?
Your problem is attempting abstract application questions like these before developing a sufficient grasp of the basic theory in a sufficient number and variety of concrete cases. You are advised to go back and review that theory.