Q17E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 15–18, find all critical points for the given system. Then use a software package to sketch the direction field in the phase plane and from this description the stability of the critical points (i.e., compare with Figure 5.12).

$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{13}\mathit{y}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{x}\mathbf{-}\mathbf{2}\mathit{y}$

Step-by-Step Solution

Answer

This is a stable node point is (0,0).

Step 1: Find critical points

Here the system is;

$$\begin{gathered} \frac{dx}{dt}=2x+13y \\ \frac{dy}{dt}=-x-2y \end{gathered}$$

For critical points equate the system equal to zero.

 $$\begin{gathered} 2x+13y=0 \\ -x-2y=0 \end{gathered}$$

Solve for x and y by eliminating the method.

 

The values of x=0 and y=0.

 

So, this is the stable node point (0,0).

Step 2: Sketch

This is the required result.