Second Order Linear Differential Equations

As before, a proof of this theorem is beyond the scope of this course.
Remark: We can solve any first order linear differential equation; Section 2-1 gives a
method for finding the general solution of any first order linear equation. In contrast, there
is no general method for solving second (or higher) order linear differential equations. There
are, however, methods for solving certain special types of second order linear equations and
we shall study these in this chapter. Extensions of these methods to higher order linear
equations will be given later.
DEFINITION 2. The linear differential equation (1) is homogeneous 1 if the function f
on the right side of the equation is 0 for all x ∈ I. In this case, equation (1) becomes
y ′′ + p(x) y ′ + q(x) y =0. (2)
Equation (1) is nonhomogeneous if f is not the zero function on I, i.e., (1) is nonhomogeneous
if f(x) = 0 for some x ∈ I.
As you will see in the work which follows, almost all of our attention will be focused on
homogeneous equations.
1
This use of the term “homogeneous” is completely different from its use to categorize the first order
equation y ′ = f(x, y) in Exercises 2.2.
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