Math Help - Verifying the solution to an ODE

I was working on solving a partial differential equation and I reduced it to an ODE, namely . I know the answer is , which works as a solution, however I'm really rusty and don't know what a good way to tackle solving the ODE is. I tried digging out my old laplace transforms tables but I'm hitting a brick wall.

Any suggestions?

Anthony

It separates.

I'm totally blind, thank you so much haha