The necessity of writing this book arose when I had to teach this course to Mechanical Engineering III year students of the PNG University of Technology, Lae. Since no consistent literature on the subject was available here, I was advised to write a text to cover the syllabus. This need, I realized earlier too, when I was Vice-Chancellor, Avadh University, Faizabad (India) in 1991 and I had to assist my daughter to learn Laplace Transforms. I looked for a suitable text but one was not available. The present book comprises seven chapters of which the first one gives a brief discussion of the essential tools for solving Ordinary Differential Equations. The next three chapters describe in detail the techniques of Laplace Transforms, Inverse Laplace Transforms and their Applications to solving Differential Equations. Fourier series have been discussed in the fifth chapter, while a short discussion of Partial Differential Equations is given in the sixth chapter; PDE''s of interest to Engineering students have been included. The last chapter offers a detailed discussion of Convergence of Infinite Series and Infinite Products.