Nonhomogeneous linear systems x’ = Ax + f(n). Variation of parameters and undetermined coefficients.

Abstract

The general nonhomogeneous linear system with constant coefficients is

$$x' = Ax + f\left( n \right)$$

(12.1)

where,as always in this chapter,A is an m×m real matrix and f: J₀ + C ^m. If f(n) = f₁(n) + if₂(n), where f₁(n) and f₂(n) are real, and if x(n) = x¹(n) + ix²(n) is a solution of (12.1), x¹(n) and x²(n) real, then ${x^l}^\prime \left( n \right) = A{x^1}\left( n \right) + {f_1}\left( n \right)$ and ${x^{2'}}\left( n \right) = A{x^2}\left( n \right) + {f_2}\left( n \right)$; and conversely, if x¹(n) and x²(n) are real solutions of ${x^{l'}} = A{x^1} + {f_1}\left( n \right)$ and ${x^{2'}} = A{x^2} + A{x^2} + {f_2}\left( n \right)$, then x(n) = x¹(n) + ix²(n) is a solution of (12.1). Thus, it is no more general to consider complex valued f(n), but it is convenient to do so. The block diagram for (12.1) is shown in Figure 12.1.