Undetermined coefficients - nonhomogenous system

• I
Heneng
Hello!

I am taking a course in differential equations and the book I am using is "Elementary Differential Equations" - E. Boyce & R. DiPrima (tenth edition)

My question is about guessing the form of a particular solution to a nonhomogenous system of equations.

Given a nonhomogenous system of differential equations on the form: x' = Px + g(t) where P is a constant n×n - matrix and g(t) is a continuous vector-function for α ∠ t ∠β.
Let υ, ν, and ρ be n×1 - vectors

On page 442 it is said:

"The procedure for choosing the form of the solution is substantially the same as that given in Section 3.5 for linear second order equations.
The main difference is illustrated by the case of a nonhomogenous term of the form υeλt , where λ is a simple root of the characteristic equation. In this situation, rather than assuming a solution of of the form νteλt , it is necessary to use νteλt + ρeλt , where ν and ρ are determined by substituting into the differential equation."

How did they come up with νteλt + ρeλt ? Why is it necessary for systems to do this and not with just a single nonhomogenous equation?

Hope someone can come up with an explanation for this.