Question

(a) Use computer software to obtain the phase portrait of the system in problem 5.if possible, include arrowheads as in figure 8.2.2also include four half lines in your phase portrait.

(b)obtain the Cartesian equations of each of the four half-lines in part(a).

(c)Draw the eigenvectors on your phase portrait of the system.

Short answer

The phase portrait of the system problem.

(a)

$x\text{'}=\begin{array}{cc}10& -5\\ 8& -12\end{array}$

b) $$\begin{gathered} x\left(t\right)=c_1\left(5 \\ 2\right)e^{8t}+c_2\left(1 \\ 4\right)e^{-10t} \end{gathered}$$

$y=4x\text{and}y=\frac{2}{5}x$

c) The eigenvectors are shown in green.

Step-by-step solution

definition of phase portrait solution

A phase portrait directional behaviour of an ordinary differential equation system

Given,

(a)

The phase portraits of the system

$x\text{'}=\begin{array}{cc}10& -5\\ 8& -12\end{array}$

Is shown below, with half-lines highlighted in red.

General solution

We previously concluded that the given system's is general solution

b)

$$\begin{gathered} x\left(t\right)=c_1\left(5 \\ 2\right)e^{8t}+c_2\left(1 \\ 4\right)e^{-10t} \end{gathered}$$

$x\left(t\right)=5c_1{e}^{8t}+c_2{e}^{-10t}$and $y\left(t\right)=2c_1{e}^{8t}+4c_2{e}^{-10t}$

We get,

$c_1=0,c_2=1$

$x=e^{-10t},y=4e^{-10t}$

Eliminating the parameter,

$y=4x,x>0$

Let,

$c_1=0,c_2=-1$

$x=e^{-10t},y=4e^{-10t}$

Eliminating the parameter

$$\begin{gathered} y=4x,c_2=-1 \\ {c}_1=1,c_2=0 \end{gathered}$$

Eliminating the parameter,

$$\begin{gathered} y=\frac{2}{5}x,x>0 \\ x=-5e^{8t} \\ y=-2e^{8t} \end{gathered}$$

Eliminating the parameter,

$y=\frac{2}{5}x,x<0$

(c) The eigenvectors are shown in green.

Hence,

Is shown below, with half-lines highlighted in red.

a)

b) $y=4x\text{and}y=\frac{2}{5}x$

c)