Question

Find phase portraits for the systems in problems $\mathbf{2}\mathbf{}\mathbf{\text{and}}\mathbf{}\mathbf{4}$for each system find any half-line trajectories and include these lines in your phase portrait.

Short answer

The Find phase portraits for the systems in problems system find any half-line trajectories and include these lines in your phase portrait.

$$\begin{gathered} X_1=\left(1 \\ 6 \\ -13\right),X_2=\left(1 \\ -2 \\ -1\right)e^{-4t},X_3=\left(2 \\ 3 \\ -2\right)e^{3t} \end{gathered}$$

$W\left(x_{1,}{x}_2,x_3\right)=\left|\begin{array}{ccc}1& 6& -13\\ e^{-4t}& -e^{-4t}& -e^{-4t}\\ 2e^{3t}& 3e^{3t}& -2e^{3t}\end{array}\right|$

Given vector is solution of system.

Step-by-step solution

definition of phase portraits

A phase portrait directional behaviour of an ordinary differential equation system (ODEs). The phase portrait can indicate the system's stability Unstable. The majority of the system's solutions tend to over time.

Given,

$$\begin{gathered} X_1=\left(1 \\ 6 \\ -13\right),X_2=\left(1 \\ -2 \\ -1\right)e^{-4t},X_3=\left(2 \\ 3 \\ -2\right)e^{3t} \end{gathered}$$

Use of determinant then,

$W\left(x_{1,}{x}_2,x_3\right)=-84e^{-t}\ne 9$

$W(x_{1,}{x}_2,x_3)=\left|\begin{array}{ccc}1& 6& -13\\ e^{-4t}& -e^{-4t}& -e^{-4t}\\ 2e^{3t}& 3e^{3t}& -2e^{3t}\end{array}\right|$

General solution of the system

The all real value of t.

We conclude that ${\mathit{x}}_{\mathbf{1}\mathbf{,}}{\mathit{x}}_{\mathbf{2}}\mathbf{,}{\mathit{x}}_{\mathbf{3}}$form a fundamental set of solution on $\left(\infty ,-\infty \right)$, Thus the general solution of the system on the interval is

$x=c_1{x}_{1,}+c_2{x}_2+c_3{x}_3$

Hence,

Given vector is solution of system.