The Laplace transform opens a new path for solving differential equations: it turns calculus problems into algebra problems, handles initial conditions seamlessly, and works beautifully with inputs that turn on, off, or switch over time. This part builds the method step by step.
The Laplace transform is one of the most powerful tools for solving differential equations—and one of the most surprising. Instead of directly working with derivatives, we temporarily reformulate the problem in a new setting called the Laplace domain. In that space, derivatives turn into algebraic terms, and the differential equation becomes something much friendlier: an algebraic equation for a new unknown, \(Y(s)\text{.}\)
The real magic is that once we solve for \(Y(s)\text{,}\) we can “come back” by applying the inverse Laplace transform and recover \(y(t)\text{,}\) the solution to the original problem. Along the way, initial conditions fold neatly into the process, and tricky features like discontinuous inputs become far easier to handle.
This chapter lays the foundation for all Laplace-related material. We’ll define the Laplace transform from scratch, explore why it works, and build a library of key transforms and properties. With those tools in hand, we’ll be ready to use the Laplace method to solve differential equations in the chapters ahead.
