Question

In problems 21-24 verify that the vector X_p given is a particular solution of the given non-homogeneous linear system.

$$\begin{gathered} \mathit{X}\mathbf{\text{'}}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{2}& \mathbf{1}\\ \mathbf{1}& \mathbf{-}\mathbf{1}\end{array}\mathbf{\right)}\mathit{X}\mathbf{+}\mathbf{\left(}\begin{array}{c}\mathbf{-}\mathbf{5}\\ \mathbf{2}\end{array}\mathbf{\right)} \\ {\mathit{X}}_{\mathbf{p}}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{3}\end{array}\mathbf{\right)} \end{gathered}$$

Short answer

Yes, the given vector $X_{\mathrm{p}}=\left(\begin{array}{c}1\\ 3\end{array}\right)$is a particular solution of the givennon-homogeneous linear system.

Step-by-step solution

Definition of general solution - Non-Homogeneous Systems

Let X_pbe a given solution of the non-homogeneous system on an interval, I,then

${\mathit{X}}_{\mathbf{c}}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathbf{X}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathbf{X}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathbf{X}}_{\mathbf{n}}$

Denote the general solution on the same interval of the homogeneous system.

Then, the general solution of the non-homogeneous system on the interval is given by,

$\mathit{X}={\mathit{X}}_c+{\mathit{X}}_p$

We are given,

$X\text{'}=\left(\begin{array}{cc}2& 1\\ 1& -1\end{array}\right)X+\left(\begin{array}{c}-5\\ 2\end{array}\right)$ ...(1)

localid="1663900419140" $X_{\mathrm{p}}=\left(\begin{array}{c}1\\ 3\end{array}\right)$ ...(2)

Obtaining the derivative of given vector

Differentiate the given vector,

$X_{\mathrm{p}}=\left[\begin{array}{c}1\\ 3\end{array}\right]$

$X_p=\left[\begin{array}{c}0\\ 0\end{array}\right]$ ...(3)

Substitute (2) and (3) in (1),

role="math" localid="1663900845840" $X_p^{\text{'}}=\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]X_p+\left[\begin{array}{c}-5\\ 2\end{array}\right]$ ...(4)

$X_p^{\text{'}}=\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]\left[\begin{array}{c}1\\ 3\end{array}\right]+\left[\begin{array}{c}-5\\ 2\end{array}\right]$

Multiply the matrices and Combine the like terms,

$$\begin{gathered} \left[\begin{array}{c}0\\ 0\end{array}\right]=\left[\begin{array}{c}2+3-5\\ 1-3+2\end{array}\right] \\ \left[\begin{array}{c}0\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right] \end{gathered}$$

i.e., we see that equation (4), $X_p^{\text{'}}=\left[\begin{array}{cc}2& 1\\ 1& -1\end{array}\right]X_p+\left[\begin{array}{c}-5\\ 2\end{array}\right]$

Thus, the given vector $X_{\mathrm{p}}=\left(\begin{array}{c}1\\ 3\end{array}\right)$is a particular solution of the givennon-homogeneous linear system.