The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. With its success; however; a certain casualness has been bred concerning its application; without much regard for hypotheses and when they are valid. Even proofs of theorems often lack rigor; and dubious mathematical practices are not uncommon in the literature for students.
In the present text; I have tried to bring to the subject a certain amount of mathematical correctness and make it accessible to undergraduates. To this end; this text addresses a number of issues that are rarely considered. For instance; when we apply the Laplace transform method to a linear ordinary differential equation with constant coefficients; any(n) + an-1y(n-1) +···+ a0y f(t); why is it justified to take the Laplace transform of both sides of the equation (Theorem A.6)? Or; in many proofs it is required to take the limit inside an integral. This is always frought with danger; especially with an improper integral; and not always justified. I have given complete details (sometimes in the Appendix) whenever this procedure is required.
Furthermore; it is sometimes desirable to take the Laplace transform of an infinite series term by term. Again it is shown that this cannot always be done; and specific sufficient conditions are established to justify this operation. Another delicate problem in the literature has been the application of the Laplace transform to the so-called Dirac delta function. Except for texts on the theory of distributions; traditional treatments are usually heuristic in nature.