Eh, I didn't check the Laplace transform so that will probably work, but here's another way: solve the general equation by looking for a solution to the homogenous and adding a particular solution. Since the funciton on the right is exponential, the simplest guess for a particular solution could be . If you plug this and equate coefficients, you should get (coulda guessed that actually) and or , so a particular solution is . The homogenous solution is so add these together and you should have a general solution.