Solving Linear Partial Differential Equations

Partial differential equations arise in many branches of science and vary in many ways. No one method can be used to solve all of them, and only a tiny percentage have been solved. In this book, we consider the general linear partial differential equation of arbitrary order m. Even this involves more methods that are known. We ask a simple when can an equation be solved and how many solutions does it have? We find that the answer is surprising even for equations with constant coefficients. We begin with these equations, first finding conditions which allow us to solve and obtain a finite number of solutions. We show how to obtain those solutions by analyzing the structure of the equation very carefully. A substantial part of the book is devoted to this. Then we tackle the more difficult problem of considering equations with variable coefficients. A large number of such equations are solved by comparing them to equations with constant coefficients. In numerous applications in the sciences, students and researchers are required to solve such equations in order to get the answers that they need. In many cases, the basic scientific theory requires that the resulting partial differential equation has a solution, and they are required to know how many solutions exist. This book deals with such situations.