Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

Second Order Linear Equations

n Chapter 3 we discussed systems of two first order equations, with primary emphasis on homogeneous linear equations with constant coefficients. In this chapter we will begin to consider second order linear equations, both homogeneous and non-homogeneous. Since second order equations can always be transformed into a system of two first order equations, this may seem redundant. However, second order equations naturally arise in many areas of application, and it is important to be able to deal with them directly. One cannot go very far in the development of fluid mechanics, heat conduction, wave motion, or electromagnetic phenomena without encountering second order linear differential equations.

4.1 Definitions and Examples

A second order differential equation is an equation involving the independent variable t, and an unknown function or dependent variable y = y(t) along with its first and second derivatives. We will assume that it is always possible to solve for the second derivative so that the equation has the form

where f is some prescribed function. Usually, we will denote the independent variable by t since time is often the independent variable ...