Q16E - 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).

$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{5}\mathit{x}\mathbf{+}\mathbf{2}\mathit{y}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{x}\mathbf{-}\mathbf{4}\mathit{y}$

Step-by-Step Solution

Answer

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

Step 1: Find critical points

Here the system is;

 $$\begin{gathered} \frac{dx}{dt}=-5x+2y \\ \frac{dy}{dt}=x-4y \end{gathered}$$ 

For critical points equate the system equal to zero.

$\begin{array}{rcl}-5x+2y& =& 0\\ x-4y& =& 0\\ & & \end{array}$ 

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.