Analytical Solutions of Ordinary Differential Equations

Abstract

In this chapter, we discuss the major approaches to obtain analytical solutions of ordinary differential equations. We begin with the solutions of first-order differential equations. Several first-order differential equations can be transformed into two major solution approaches: the separation of variables approach and the exact differential approach. We start with a brief review of both approaches, and then we follow them with two sections on how to reduce other problems to either of these methods. First, we discuss the use of similarity transformations to reduce differential equations to become separable. We show that these transformations cover other well-known approaches, such as homogeneous-type differential equations and isobaric differential equations, as special cases. The next section continues with the search for integrating factors that would transform a given differential equation to become exact. Important special cases of this approach include first-order linear differential equations and the Bernoulli equations (after some additional variable transformation). Next, we discuss the solution of second-order differential equations. We opted to focus first on the nonlinear types, leaving the solution of linear second-order differential equations to be included in the later sections that handle high-order linear differential equations. The approaches we consider are those that would reduce the order of the differential equations, with the expectation that once they are first-order equations, techniques of the previous sections can be used to continue the solution process. Specifically, we use a change of variables to handle the cases in which either the independent variable or dependent variable are explicitly absent in the differential equation.