Differential Equations - Extra Credit Induction Proof Involving Laplace Transf.

This is an extra credit assignment I use for my Differential Equation class. It involves a very wild Laplace transform that is actually tied to the Gamma Function. I do not ask students to see that connection, rather I expect them to be able to prove this transform holds using Induction. The problem is very complex and deep, it is a really fun proof and uses some nontrivial Calculus III along the way too. Attached is the problem statement and a very thorough multipage solution.

The assignment comes with detailed solutions in the same pdf so you may have to use a free website like pdfescape to split the solutions from the homework assignment.