Vtyjkyuk

This is actually essentially a question on what to do with the constants. I'm trying to derive the underdamped solution for a spring.

For my trial solution $x(t) = Aexpleft(atright)$, I have

$$a = -Gamma pm omega$$

$$therefore x(t) = Aexp(left(-Gamma + omega)tright) + Bexp(left(-Gamma - omega)tright)$$

Anyway, I eventually get to (and I'm pretty sure my math is right):

$$x(t) = exp(-Gamma t) left ((A+B)cos(omega t) + i(A-B)sin(omega t)right)$$

So, if I let $A+B = C$, that looks nicer.

However, I feel tentative about just saying $$i(A-B) = D$$ without explicitly saying that $D$ is complex, and even then I don't think that's right. I don't think there's supposed to be a complex part to this solution - so what do I do? How do I turn this into a real constant?