Use the Laplace transform to solve the following initial value problem:
First, using for the Laplace transform of , i.e., , find the equation you get by taking the Laplace transform of the differential equation and solving for :

Find the partial fraction decomposition of and its inverse Laplace transform to find the solution of the DE:

Use the Laplace transform to solve the following initial value problem:
Let , and
. Find the expressions you obtain by taking the Laplace transform of both differential equations and solving for and :

Find the partial fraction decomposition of and and their inverse Laplace transforms to find the solution of the system of DEs: