Differential Equations and Linear Algebra, 4th Edition

5.3 Homogeneous Equations with Constant Coefficients

In Section 5.2 we saw that a general solution of an nth-order homogeneous linear equation is a linear combination of n linearly independent particular solutions, but we said little about how actually to find even a single solution. The solution of a linear differential equation with variable coefficients ordinarily requires numerical methods (Chapter 2) or infinite series methods (Chapter 8). But we can now show how to find, explicitly and in a rather straightforward way, n linearly independent solutions of a given nth-order linear equation if it has constant coefficients. The general such equation may be written in the form

$$a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_{2}y″+a_{1}y\text{′}+a_{0}y=0,$$ (1)

where the coefficients ...