# Deriving the second order differential equation general solution with complex roots

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 $x(t) = A\exp{\left(at\right)}$, I have

$$a = -\Gamma \pm \omega$$

$$\therefore \ x(t) = A\exp{(\left(-\Gamma + \omega)t\right)} + B\exp{(\left(-\Gamma - \omega)t\right)}$$

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?