How to deal with duplication in the method of undetermined coefficients in systems of nonhomogeneous ODEs? I have the following problem and its solution from the book. What I don't understand is- why do we still have $b_1e^{-t}$ and $b_2e^{-t}$ in $x_p$ and $y_p$ since it can still cause duplication? My understanding is that $x_p$ should be $a_1+b_1te^{-t}$ and similar for $y_p.$ I tried it but I got some inconsistency. Fyi; I am using this table to make an educated guess for the form of particular solution that I should pick and how to deal with duplication.
 A: The method of undetermined coefficients works when the forcing terms and the trial solutions belong to a class of functions that is left unchanged under differentiation. For example, any function of the form $z(t) = A\cos(2t) + B\sin(2t)$ stays as a function of the form $A\cos(2t) + B\sin(2t)$ when you differentiate in $t$, with the constants $A$ and $B$ the only things that change. This is necessary for you to be able to match all the coefficients.
The issue with taking $x_p(t) = A + Bte^{-t}$ is that the class of all functions of this form is not invariant under differentiation. We see that taking one derivative results in $x_p'(t) = Be^{-t} - Bte^{-t}$, which belongs to the class of functions of the form $A + Be^{-t} + Cte^{-t}$: so we see that we get an $e^{-t}$ term that was not present before.
Since $x' = 6x - 7y + 10$, the LHS is a function of the form $A + Be^{-t} + Cte^{-t}$. If both $x_p$ and $y_p$ are of the form $C + Dte^{-t}$ then the RHS is also a function of the form $C + Dte^{-t}$. So you cannot match coefficients unless $B=0$, and this redundancy ultimately results in the inconsistencies you're coming across. The same can be said for the $y'$ equation Introducing the $e^{-t}$ terms into the trial solutions avoids this problem: both the LHS and RHS become functions of the same form.
