Outline of procedure: Use the Laplace Transform (LT) in the time domain only. Result: second-order ODE in C(x,s). Solve using standard methods. Result: C(x,s). Take the inverse LT of this expression to obtain c(x,t).
Warning: the inverse LT can easily become quite hairy here, since you may not be able to find the inverse LT in a table. You might have to go back to the definition of the inverse LT in terms of a complex line integral, compute residues, etc. It's entirely possible that your solution will turn out to be an infinite series solution. That's typical.
Does this make sense?