First, I don't see any C in the equation to solve, so I assume it's the constant of integration.
I got:
\frac{\sqrt{B}}{\sqrt{k}}\tanh{\frac{\sqrt{B}\sqrt{k}\cdot t+C \sqrt{B}\sqrt{k}\cdot m}{m}}
I couldn't read trough all those symbols, but from what I gather, you're asked to Laplace trasform those equations by definition. Try to look at a Laplace transform book and you're done