The Problem

Solve the following ordinary differential equation

using the method of Laplace transforms.

The Solution

If you need to brush up your knowledge on Laplace transforms, you may want to read this article, first:

First, we perform a Laplace transform of the differential equation:

where 𝑌 is the Laplace transform of 𝑦(𝑡) and we have made use of the fact that the Laplace transform is linear. We also used the shifting theorem and the basic Laplace transform of 1 on the right-hand side.

Next, we insert the initial conditions and solve for 𝑌:

Transforming back is easy, since (up to a parameter shift) each term is one of the basic Laplace transforms everyone should know: