6.39 Laplace transforms

Recall the definition of the Laplace transform from slide 1.59:

The main application of Laplace transforms is to the solution of differential equations. The idea is that taking Laplace transforms converts a differential equation into an algebraic equation, which we can solve; to obtain a solution to the original differential equation we need to apply an ‘inverse Laplace transform’ to our solution.

We denote the Laplace transform of $f(x)$ by ${\mathcal{L}}(f)(s)$, or just ${\mathcal{L}}(f)$.