Differential Equations and Linear Algebra, 4th Edition

8.2 Nonhomogeneous Linear Systems

In Section 5.5 we exhibited two techniques for finding a single particular solution of a single nonhomogeneous nth-order linear differential equation—the method of undetermined coefficients and the method of variation of parameters. Each of these may be generalized to nonhomogeneous linear systems. In a linear system modeling a physical situation, nonhomogeneous terms typically correspond to external influences, such as the inflow of liquid to a cascade of brine tanks or an external force acting on a mass-and-spring system.

Given the nonhomogeneous first-order linear system

$${\mathbf{\text{x}}}^{\text{′}}=\mathbf{\text{Ax}}+\mathbf{\text{f}}(t)$$ (1)

where A is an $n\times n$ constant ...