where L[u(x,t)] = U(x,s) with respect to t

Given u(x,o)=x u(0.t)=o

It is an imhonogeous equation. It need to find out general soland particular sol.

if it is a honogeous equation, just very easy to do.Am I correct in finding general sol.?

what is the yp(particular sol) in order to find U(x,s)

Finally, How to do inverse laplace transform to find u(x,t).

Thank