Chapter 10 Backward Transforms
Applying the Laplace transform method to a differential equation starts by transforming this equation into a new equation where the derivatives are replaced by algebraic expressions. To recover to the solution to the original equation the transformation must also be reversed. This process, which we will refer to as a “backwards transform”, corresponds to step 3 on the Laplace transform roadmap.
Just as the forward Laplace transform simplifies a differential equation by converting it into an algebraic one, the backward transform completes the cycle by converting the algebraic solution back into its original form. This chapter will focus on the techniques used to perform the inverse Laplace transform, guiding you through the steps needed to move from the transformed function, \(F(s)\text{,}\) back to the original function, \(f(t)\text{.}\)
We will explore key methods, such as partial fraction decomposition and the use of Laplace transform tables, to help simplify complex transformed functions. Additionally, we will highlight common challenges and how to overcome them, ensuring a smooth transition back to the original domain.
