91影视

Let the equations from the given determinant are as follows:

$$\begin{gathered} \frac{\mathbf{d}\mathbf{胃}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{蝇} \\ \frac{\mathbf{d}\mathbf{蝇}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{p}\mathit{胃}\mathbf{-}\mathit{q}\mathit{蝇} \end{gathered}$$

The corresponding matrix for the system is:

$A=\left[\begin{array}{cc}0& 1\\ -p& -q\end{array}\right]$

Eigen values are given by the characteristic equation

$\begin{array}{rcl}\left|A-位I\right|& =& 0\\ \left|\begin{array}{cc}-位& 1\\ -p& -q-位\end{array}\right|& =& 0\\ {位}^2+q位+p& =& 0\\ 位& =& \frac{-q卤\sqrt{q^2-4q}}{2},q^2>4q\end{array}$

Thus, the Eigen values of are negative.

Hence, zero state is a stable equilibrium.

The phase portrait is as follows:

Let $E_{{位}_1}$, $E_{{位}_2}$are Eigen spaces of ${位}_1$and role="math" localid="1665036517865" ${位}_2$. Since both eigenvalues are negative.

Hence, the solution is:

$\stackrel{鈫�}{x}\left(t\right)=C_1{e}^{{位}_{1}t}\stackrel{鈫�}{v_1}+C_2{e}^{{位}_{2}t}\stackrel{鈫�}{v_2}$

Here, the values are $\stackrel{鈫�}{x}=\left[\begin{array}{c}胃\\ 蝇\end{array}\right]$, role="math" localid="1665035976107" $\stackrel{鈫�}{v_1}$corresponding to ${位}_1=\frac{-q-\sqrt{q^2-4q}}{2}$and $\stackrel{鈫�}{v_2}$ corresponding to ${位}_2=\frac{-q+\sqrt{q^2-4q}}{2}$.

As $t鈫�鈭�,\stackrel{鈫�}{x}\left(t\right)鈫�0$solution converges to origin.

Also, $C_1$and$C_2$ are arbitrary constants.

Dominant terms is $C_2{e}^{{位}_{2}t}\stackrel{鈫�}{v_2}$as ${位}_2$is larger in magnitude, therefore distant trajectories are parallel to $E_{{位}_2}$.

All trajectories converge to origin.Along the trajectory 1, the movement of the door slow down continuously until it reaches to the closed state as along this trajectory w decreases continuously.

Door slams if the trajectory 2 is followed that is initially if :

$\frac{蝇\left(0\right)}{胃\left(0\right)}<{位}_2=\frac{-q\sqrt{q^2+4q}}{2}$

From figure it observe that for condition $蝇<0$door reaches$胃<0$ .

Hence, all trajectories converge to origin and it observe that for condition $蝇<0$door reaches$胃<0$ .