Question

Verify that the given vector X is a solution of the given homogenous linear equation;

$$\begin{gathered} \mathit{X}\mathbf{\text{'}}\mathbf{=}\mathbf{\left(}\begin{array}{ccc}\mathbf{1}& \mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{1}& \mathbf{0}\\ \mathbf{-}\mathbf{2}& \mathbf{0}& \mathbf{-}\mathbf{1}\end{array}\mathbf{\right)}\mathit{X}\mathbf{;} \\ \mathbf{}\mathit{X}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{sint}\\ \mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{sint}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{cost}\\ \mathbf{-}\mathbf{sint}\mathbf{+}\mathbf{cost}\end{array}\mathbf{\right)} \end{gathered}$$

Short answer

Vector X is verified, it is the solution of given homogenous linear system:

$\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ -2& 0& -1\end{array}\right)X=\left(\begin{array}{c}\cos t\\ -\frac{1}{2}\cos t+\frac{1}{2}\sin t\\ -\cos t-\sin t\end{array}\right)$

Step-by-step solution

Homogeneous linear equation;

Let ${\mathit{X}}_{\mathbf{1}}\mathbf{,}{\mathit{X}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\mathit{X}}_{\mathbf{n}}$be a fundamental set of solutions of homogeneous system on an interval l. Then the general solution of the system on the intervals is

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

Where the ${\mathit{c}}_{\mathbf{i}}\mathbf{,}\mathit{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{n}$ are arbitrary constant.

Verification of the vector:

Here we will verify that the given vector X is the solution of the given homogenous linear system;

$$\begin{gathered} X\text{'}=\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ -2& 0& -1\end{array}\right)X; \\ X=\left(\begin{array}{c}\mathrm{sint}\\ -\frac{1}{2}\mathrm{sint}-\frac{1}{2}\mathrm{cost}\\ -\mathrm{sint}+\mathrm{cost}\end{array}\right) \\ X\text{'}=\left(\begin{array}{c}\left(\mathrm{sint}\right)\text{'}\\ {\left(-\frac{1}{2}\mathrm{sint}-\frac{1}{2}\mathrm{cost}\right)}^{\text{'}}\\ \left(-\mathrm{sint}+\mathrm{cost}\right)\text{'}\end{array}\right) \\ X\text{'}=\left(\begin{array}{c}\cos t\\ -\frac{1}{2}\cos t+\frac{1}{2}\sin t\\ -\cos t-\sin t\end{array}\right) \end{gathered}$$

$$\begin{gathered} \left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ -2& 0& -1\end{array}\right)X=\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ -2& 0& -1\end{array}\right)\left(\begin{array}{c}\mathrm{sint}\\ -\frac{1}{2}\mathrm{sint}-\frac{1}{2}\mathrm{cost}\\ -\mathrm{sint}+\mathrm{cost}\end{array}\right) \\ \left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ -2& 0& -1\end{array}\right)X=\left(\begin{array}{c}\cos t\\ -\frac{1}{2}\cos t+\frac{1}{2}\sin t\\ -\cos t-\sin t\end{array}\right) \end{gathered}$$

Hence, the vector X is verified:$\left(\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ -2& 0& -1\end{array}\right)X=\left(\begin{array}{c}\cos t\\ -\frac{1}{2}\cos t+\frac{1}{2}\sin t\\ -\cos t-\sin t\end{array}\right)$