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

To begin our exploration of the Laplace Transform, we start with the first step in the Laplace Transform Roadmap, known as the forward Laplace transform. This step is foundational, as it sets the stage for the subsequent transformations outlined in the roadmap.

Applying the forward Laplace transform to a differential equation involves converting both the left-hand side (LHS) and right-hand side (RHS) of the equation into their Laplace-transformed counterparts.

This section investigates how individual functions, like $-40e^{3t}$ in step 1, are transformed into expressions such as ${\displaystyle -\frac{40}{s-3}\text{.}}$ In the next section, we will show where the $s^{2}Y+3sY+2Y-1$ comes from in the forward transform.