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CHAPTER 3
Second Order Linear Differential Equations
3.1 Introduction; Basic Terminology and Results
Any second order differential equation can be written as
F (x, y, y′, y′′) = 0
This chapter is concerned with special yet very important second order equations, namely linear equations.
Recall that a first order linear differential equation is an equation which can be written in the form y′^ + p(x)y = q(x)
where p and q are continuous functions on some interval I. A second order, linear differential equation has an analogous form.
DEFINITION 1. A second order linear differential equation is an equation which can be written in the form
y′′^ + p(x)y′^ + q(x)y = f (x) (1)
where p, q, and f are continuous functions on some interval I.
The functions p and q are called the coefficients of the equation; the function f on the right-hand side is called the forcing function or the nonhomogeneous term. The term “forcing function” comes from applications of second-order linear equations; the description “nonhomogeneous” is given below.
A second order equation which is not linear is said to be nonlinear.
Examples
(a) y′′^ − 5 y′^ + 6y = 3 cos 2x. Here p(x) = − 5 , q(x) = 6, f (x) = 3 cos 2x are continuous functions on (−∞, ∞).
(b) x^2 y′′^ − 2 x y′^ + 2y = 0. This equation is linear because it can be written in the form (1) as y′′^ − 2 x
y′^ +^2 x^2
y = 0
where p(x) = 2/x, q(x) = 2/x^2 , f (x) = 0 are continuous on any interval that does not contain x = 0. For example, we could take I = (0, ∞).
(c) y′′^ + xy^2 y′^ − y^3 = exy^ is a nonlinear equation; this equation cannot be written in the form (1).
Remarks on “Linear.” Intuitively, a second order differential equation is linear if y′′ appears in the equation with exponent 1 only, and if either or both of y and y′^ appear in the equation, then they do so with exponent 1 only. Also, there are no so-called “cross- product” terms, y y′, y y′′, y′^ y′′. In this sense, it is easy to see that the equations in (a) and (b) are linear, and the equation in (c) is nonlinear.
Set L[y] = y′′^ + p(x)y′^ + q(x)y. If we view L as an “operator” that transforms a twice differentiable function y = y(x) into the continuous function
L[y(x)] = y′′(x) + p(x)y′(x) + q(x)y(x),
then, for any two twice differentiable functions y 1 (x) and y 2 (x),
L[y 1 (x) + y 2 (x)] = [y 1 (x) + y 2 (x)]′′^ + p(x)[y 1 (x) + y 2 (x)]′^ + q(x)[y 1 (x) + y 2 (x)]
= y 1 ′′ (x) + y 2 ′′ (x) + p(x)[y′ 1 (x) + y′ 2 (x)] + q(x)[y 1 (x) + y 2 (x)]
= y 1 ′′ (x) + p(x)y′ 1 (x) + q(x)y 1 (x) + y′′ 2 (x) + p(x)y′ 2 (x) + q(x) + y 2 (x)
= L[y 1 (x)] + L[y 2 (x)]
and, for any constant c,
L[cy(x)] = [cy(x)]′′^ + p(x)[cy(x)]′^ + q(x)[cy(x)]
= cy′′(x) + p(x)[cy′(x)] + cq(x)y(x)
= c[y′′(x) + p(x)y′(x) + q(x)y(x)]
= cL[y(x)].
Therefore, as introduced in Section 2.1, L is a linear differential operator. This is the real reason that equation (1) is said to be a linear differential equation.
The first thing we need to know is that an initial-value problem has a solution, and that it is unique.
THEOREM 1. (Existence and Uniqueness Theorem) Given the second order linear equation (1). Let a be any point on the interval I, and let α and β be any two real numbers. Then the initial-value problem
y′′^ + p(x) y′^ + q(x) y = f (x), y(a) = α, y′(a) = β
has a unique solution.
