## ABSTRACT

In a previous chapter, it was seen that a fundamental set of solutions for any second-order homogeneous linear equation with constant coefficients can be found in terms of exponential functions whose exponents are given by the roots of a corresponding second-degree polynomial equation—the corresponding “characteristic equation”. Unsurprisingly, the same basic approach can be used to find a fundamental set of solutions for any homogeneous linear equation with constant coefficients, no matter what the order. A major complication, of course, is the added algebra, since the major tool for solving arbitrary second-degree polynomial equations, the quadratic formula, does not extend to a corresponding tool for solving arbitrary higher-degree polynomial equations.

In this chapter, the basic theory and methods for second-order equations are extended to handle constant-coefficient homogeneous linear \des of any order. Of particular consideration are those chases in which a root of the characteristic polynomial has a multiplicity greater than one, especially when that root is complex. In addition, practical advice on finding the roots of higher-degree polynomial equations is given along with several examples.