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

The general nonhomogeneous linear system with constant coefficients is 12.1 $$x' = Ax + f\left( n \right)$$ where,as always in this chapter,A is an m×m real matrix and f: J₀ + Cm. 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.