ABSTRACT
An extremely important class of second-order homogeneous linear differential equations are those in which the coefficients are constants, that is, those that can be written as https://www.w3.org/1998/Math/MathML"> a y ″ + b y ′ + c y = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429347429/d64fdc95-3047-445f-b6ba-9acf0bfd534d/content/eq2520.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where a, b and c are real-valued constants with a ≠ 0. These equations are important for two reasons: One is that they often arise in applications. The other is that, as shown in this chapter, they are relatively easy to solve.
The chapter begins by showing why exponential solutions should be expected, and observing that, if y = erx , then the above second-order differential equation reduces to a simple second-degree polynomial equation, the “characteristic equation”, https://www.w3.org/1998/Math/MathML"> a r 2 + b r + c = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429347429/d64fdc95-3047-445f-b6ba-9acf0bfd534d/content/eq2521.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Applying the quadratic formula leads to three different cases: a distinct pair of real values for r, a single real value for r and a conjugate pair of complex values for r. The rest of the chapter is then devoted to deriving a corresponding fundamental set of real-valued solutions to the original differential equation.
