The Non-Homogeneous Linear Equation

Abstract

Abstract In this chapter we shall consider the non-homogeneous second-order linear equation where f : [a, b] → ℝ and each p i: [a, b] → ℝ are continuous, and p 2(x) > 0, for each x in [a, b]. Any particular solution of (1) (or, indeed, of any differential equation) is called a particular integral of the equation, whereas the general solution c 1 y 1 + c 2 y 2 of the corresponding homogeneous equation given by Proposition 6 of Chapter 3 (where y 1, y 2 are linearly independent solutions of (2) and c 1, c 2 are arbitrary real constants) is called the complementary function of (1). If y P is any particular integral and y C denotes the complementary function of (1), y C + y P is called the general solution of (1). We justify this terminology by the following proposition, which shows that, once two linearly independent solutions of (2) are found, all that remains to be done in solving (1) is to find one particular integral.