Q18E - 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).

$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{x}\mathbf{(}\mathbf{7}\mathbf{-}\mathit{x}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{)}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{y}\mathbf{(}\mathbf{5}\mathbf{-}\mathit{x}\mathbf{-}\mathit{y}\mathbf{)}$

Step-by-Step Solution

Answer

This result is the unstable node point (0,0),stable node (7,0),stable node (0,5) and saddle point (3,2).

Step 1: Find critical points

Here the system is;

$$\begin{gathered} \frac{dx}{dt}=x(7-x-2y) \\ \frac{dy}{dt}=y(5-x-y) \end{gathered}$$ 

For critical points equate the system equal to zero.

$$\begin{gathered} x(7-x-2y)=0 \\ y(5-x-y)=0 \end{gathered}$$ 

Solve for x and y get the four points (0,0),(7,0),(0,5),(3,2).

 

So, this is the unstable node point (0, 0), stable node (7, 0), stable node (0, 5) and saddle point (3, 2).

Step 2: Sketch

This is the required result.