# Damped Oscillator probelm

Here's the problem:
A damped oscillator has a mass of .05 kg, a spring constant of 5 N/m, and a damping constant of .4 Ns/m. At t=0, the mass is moving at 3.0 m/s at x=.1m. Find x as a function of time.

What I have done:
I know the damping constant b = .4 and I have used this to find Beta. Also, I used k and m to find w0. I know the general solution for the equation of the harmonic oscillator -- please pardon my typing -- x(t) = e^(-beta*t)[A1* e^(sqrt (Beta^2-w0^2)) *t + A2 *e^(-sqrt(Beta^2-w0^2))].
I can use my initial condition for the position to get one equation with A1 and A2 in it. I can then take the derivative of x(t) and use the initial condition of the velocity to get the other. I now have two equations and two unknowns. The issue is that these unknowns will involve imaginary numbers because we have an underdamped case. Is this ok?

Yes, it is OK. You should find that the imaginary parts of $A_1$ and $A_2$ are exactly cancelled out by the imaginary parts of the complex exponentials to give you a real function.
Remember, to be real an expression doesn't have to have a complete absence of the imaginary unit $i$. It simply has to equal its own complex conjugate. For instance the expression $(x-3i)(x+3i)$ is purely real, despite the fact that $i$ appears.