Chapter 6.6, Problem 14P

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

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

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

Since the entries of $g\left(t\right)$ are linear functions of $t$, substitute an expression of the form

${\text{x}}_p\left(t\right)=ta+b=t\left(\begin{array}{l}{a}_1\\ a_2\end{array}\right)+\left(\begin{array}{l}{b}_1\\ b_2\end{array}\right)$

Into Eq. (ii) and match the coefficients of the vector polynomial on both sides of the resulting equation to obtain the two systems

$\text{Aa}=-\left(\begin{array}{c}0\\ 3\end{array}\right),\text{Ab}\text{=}\text{a}$.

Solve the first equation for $\text{a}$, substitute the result into the second equation, and solve for $\text{b}$.