Please help...
Use the Laplace transform and solve the following initial value problem:
d^{2}y/dt^{2 }– 4 dy/dt +4y = t^{3}e^{2t }; y(0) = 0,dy/dt|_{t =0} = 0
If you are new to it, it's probably a good idea to read your textbook, or whatever you are learning from, to learn the steps to using the Laplace transform for solving differential equations.
Asking us to solve it for you when you have no idea what you are doing isn't going to help you any.
Great! So show what you tried so we will know where you are having difficulty. Do you know what the "Laplace transform" is? If so do you know how to find the Laplace transform of $\displaystyle dy/dt$ and $\displaystyle d^2y/dt^2$? Have you found the Laplace transform of both sides of this equation?