Interactive Differential Equations: A Step-by-Step Approach to Methods & Modeling

The Laplace Transform Method provides a powerful toolset for solving initial-value differential equations, transforming complex calculus problems into simpler algebraic ones. In the previous sections, we explored how to compute the forward Laplace transform of a function and how to apply the inverse Laplace transform to return to the time domain. These two operations are the fundamental components of the Laplace Transform Method.

This section brings together the forward and backward transformations to solve entire initial-value differential equations. By applying the forward Laplace transform to both sides of a differential equation, we convert the problem into an algebraic equation in the $s$-domain. The algebraic equation is typically easier to solve, and once we find the solution in the $s$-domain, we apply the inverse Laplace transform to recover the solution in the time domain.

We will begin with simple examples and progressively tackle more complex problems, incorporating the initial conditions of the differential equations directly into the transformed equations. By mastering these techniques, you will be able to solve a wide range of differential equations using the Laplace Transform Method.