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THE LINEAR SECOND-ORDER EQUATION
THE CONSTANT COEFFICIENT CASE THE
CHAPTER 2 NONHOMOGENEOUS EQUATION SPRING
MOTION EULER’S DIFFERENTIAL EQUATION
Linear

Second-Order

Equations

A second-order differential equation is one containing a second derivative but no higher deriva-
tive. The theory of second-order differential equations is vast, and we will focus on linear
second-order equations, which have many important uses.

2.1 The Linear Second-Order Equation

This section lays the foundations for writing solutions of the second-order linear differential
equation. Generally, this equation is

P(x)y + Q(x)y (x) + R(x)y(x) = F(x).

Notice that this equation “loses” its second derivative at any point where P(x) is zero, presenting
technical difficulties in writing solutions. We will therefore begin by restricting the equation to
intervals (perhaps the entire real line) on which P(x) = 0. On such an interval, we can divide the
differential equation by P(x) and confine our attention to the important case

y + p(x)y + q(x)y = f (x). (2.1)

We will refer to this as the second-order linear differential equation.
Often, we assume that p and q are continuous (at least on the interval where we seek solu-
tions). The function f is called a forcing function for the differential equation, and in some
applications, it can have finitely many jump discontinuities.
To get some feeling for what we are dealing with, consider a simple example

y − 12x = 0.
Since y = 12x, we can integrate once to obtain

2
y (x) = 12xdx = 6x + c
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