Question

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

$$\begin{gathered} \frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{5}\mathit{y} \\ \frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{;} \end{gathered}$$ $\mathit{X}\mathbf{=}\mathbf{\left(}\begin{array}{c}\mathbf{5}\mathbf{cost}\\ \mathbf{3}\mathbf{cost}\mathbf{-}\mathbf{sint}\end{array}\mathbf{\right)}{\mathit{e}}^{\mathbf{t}}$

Short answer

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

$\therefore AX=X\text{'}$

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;

$A=\left(\begin{array}{cc}-2& 5\\ -2& 4\end{array}\right)$

The product of A and X is:

$$\begin{gathered} =\left(\begin{array}{cc}-2& 5\\ -2& 4\end{array}\right)\left(\begin{array}{c}5\cos t\\ 3\cos t-\sin t\end{array}\right)e^t \\ =\left(\begin{array}{c}-10\cos t+5(3\cos t-\sin t)\\ -10\cos t+4(3\cos t-\sin t)\end{array}\right)e^t \\ =\left(\begin{array}{c}5\cos t-5\sin t\\ 2\cos t-4\sin t\end{array}\right)e^t \\ =\left(\begin{array}{c}5e^t\left(\cos \right(t)-\sin (t\left)\right)\\ -4e^t\sin \left(t\right)+2e^t\cos \left(t\right)\end{array}\right) \end{gathered}$$

role="math" localid="1663878782307" $\therefore AX=X\text{'}$

Hence, the vector X is verified:$AX=X\text{'}$