10.1. Finding Laplace Transforms
To unlock the magic of Laplace transforms, you need to be able to find the Laplace transform of the differential equation you're working with. That's why this section gives you practice calculating Laplace transforms of various mathematical expressions, such as exponentials and trigonometry functions.
Here's what a general integral transform looks like (note that this transform is not yet a Laplace transform):
In this case, f(t) is the function you're taking an integral transform of, and F(s) is the transform. K(s, t) is called the kernel of the transform, which is the function you mix into the integral. (When calculating a Laplace transform, you choose your own kernel because doing so gives you a chance to simplify your differential equation.) The limits of integration, α and β, can be anything you choose, but the most common limits for Laplace transforms are 0 to + ∞.
To calculate a Laplace transform by hand, simply follow these steps:
Choose a kernel that transforms a differential equation into something simpler.
Try to invert the transform to get the solution of your original differential equation.