Chapter 6.6, Problem 15P

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

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

$g\left(t\right)=\left(\begin{array}{l}\sin t\\ 0\end{array}\right)=\sin t\left(\begin{array}{l}1\\ 0\end{array}\right)$.

Since the entry of $g\left(t\right)$ contains a sine function, substitute an expression of the form

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

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

$\text{Aa}=\text{b},\text{Ab}\text{=}-\text{a}-\left(\begin{array}{c}1\\ 0\end{array}\right)$.

Show that $\left({\text{A}}^{\text{2}}{\text{+I}}_{\text{2}}\right)\text{a}=-\left(\begin{array}{c}1\\ 0\end{array}\right)$ and solvefor $\text{a}$. Then substitute thisresult into the second equation and solve for $\text{b}$.