How Do You Solve for X(t) in an Underdamped Oscillator Problem?

[don_anon25]

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?

 

[quantumdude]

Yes, it is OK. You should find that the imaginary parts of $A_1$ and $A_2$ are exactly canceled 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.