Chapter 6.6, Problem 16P

Undetermined Coefficients. For each of the nonhomogeneous terms specified in Problems 14 through 16, use the method of undetermined coefficients to find a particular solution of

$\text{x}\text{'}=\left(\begin{array}{cc}-2& 1\\ 1& -2\end{array}\right)\text{x}+g\left(t\right)=\text{Ax}+g\left(t\right)$ (ii)

Given that the general solution of the corresponding homogeneous system $\text{x'}\text{=}\text{Ax}$ is

$\text{x}=c_1{e}^{-3t}\left(\begin{array}{c}1\\ -1\end{array}\right)+c_2{e}^{-t}\left(\begin{array}{c}1\\ 1\end{array}\right)$

$g\left(t\right)=\left(\begin{array}{l}6e^{-t}\\ 0\end{array}\right)=e^{-t}\left(\begin{array}{l}6\\ 0\end{array}\right)$.

Since the homogeneous equation has a solution of the form ${\text{x}}_p\left(t\right)=e^{-t}\text{a}$, it is necessary to include both $t{e}^{-t}\text{a}$ and $e^{-t}\text{b}$ in the assumed form for the particular solution,

${\text{x}}_p\left(t\right)=e^{-t}\text{b+}t{e}^{-t}\text{a}$.

Substitute this expression into Eq. (ii) using the given $g$ and show that $\text{a}$ and $\text{b}$ must be selected to satisfy

$\left({\text{A+I}}_{\text{2}}\right)\text{a}=0,\left({\text{A+I}}_{\text{2}}\right)\text{b}\text{=}\text{a}-\left(\begin{array}{c}2\\ 0\end{array}\right)$.

Then find $\text{a}$ and $\text{b}$ that satisfy both of these equations.