Hello, I am having great difficulty with the following problem:

where for and for . Initial values are and .

I wrote as and used Laplace transforms to get

.

Originally I distributed the , solved for and ended up with some crazy partial fractions, but I think this definitely can't be right as I'm supposed to end up with another piecewise function as my answer. Could anyone point me in the right direction? I have never done a piecewise function with functions of t as the values.