Question

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

$$\begin{gathered} \mathbf{}\mathit{X}\mathbf{\text{'}}\mathbf{=}\mathbf{\left(}\begin{array}{cc}\mathbf{2}& \mathbf{1}\\ \mathbf{3}& \mathbf{4}\end{array}\mathbf{\right)}\mathit{X}\mathbf{-}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{7}\end{array}\mathbf{\right)}{\mathit{e}}^{\mathbf{t}} \\ {\mathit{X}}_{\mathbf{p}}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{1}\end{array}\mathbf{\right)}{\mathit{e}}^{\mathbf{t}}\mathbf{+}\mathbf{\left(}\begin{array}{c}\mathbf{1}\\ \mathbf{-}\mathbf{1}\end{array}\mathbf{\right)}\mathit{t}{\mathit{e}}^{\mathbf{t}} \end{gathered}$$

Short answer

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

Step-by-step solution

Definition of general solution - Non-Homogeneous Systems

Let X_p be 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,

$X\text{'}=\left(\begin{array}{cc}2& 1\\ 3& 4\end{array}\right)X-\left(\begin{array}{c}1\\ 7\end{array}\right)e^{\mathrm{t}}$ ...(1)

role="math" localid="1663901799932" $X_{\mathrm{p}}=\left(\begin{array}{c}1\\ 1\end{array}\right)e^{\mathrm{t}}+\left(\begin{array}{c}1\\ -1\end{array}\right)t{e}^{\mathrm{t}}$ ...(2)

Obtaining the derivative of given vector

Differentiate the given vector,

$X_{\mathrm{p}}=\left[\begin{array}{c}1\\ 1\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}$

$X_p^{\text{'}}=\left[\begin{array}{c}2\\ 0\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}$ ...(3)

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

$X_p^{\text{'}}=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]X_p-\left[\begin{array}{c}1\\ 7\end{array}\right]e^t$ ...(4)

$\left[\begin{array}{c}2\\ 0\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]\left(\left[\begin{array}{c}1\\ 1\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}\right)-\left[\begin{array}{c}1\\ 7\end{array}\right]e^{\mathrm{t}}$

Multiply the matrices and Combine the like terms,

$$\begin{gathered} \left[\begin{array}{c}2\\ 0\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}=\left[\begin{array}{c}2+1\\ 3+4\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}2-1\\ 3-4\end{array}\right]t{e}^{\mathrm{t}}-\left[\begin{array}{c}1\\ 7\end{array}\right]e^{\mathrm{t}} \\ \left[\begin{array}{c}2\\ 0\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}=\left[\begin{array}{c}3\\ 7\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}-\left[\begin{array}{c}1\\ 7\end{array}\right]e^{\mathrm{t}} \\ \left[\begin{array}{c}2\\ 0\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}}=\left[\begin{array}{c}2\\ 0\end{array}\right]e^{\mathrm{t}}+\left[\begin{array}{c}1\\ -1\end{array}\right]t{e}^{\mathrm{t}} \end{gathered}$$

i.e., we see that equation (4), $X_p^{\text{'}}=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]X_p-\left[\begin{array}{c}1\\ 7\end{array}\right]e^t$

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