Q4E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 3–6, find the critical point set for the given system.

$$\begin{gathered} \frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{y}\mathbf{-}\mathbf{1} \\ \frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{x}\mathbf{+}\mathit{y}\mathbf{+}\mathbf{5} \end{gathered}$$

Step-by-Step Solution

Answer

The only critical points are (-6,1).

Step 1: Find critical points

Consider the system as;

 $$\begin{gathered} \frac{dx}{dt}=y-1 \\ \frac{dy}{dt}=x+y+5 \end{gathered}$$

 

For finding the critical points put the system equal to 0. 

So, 

$\begin{array}{rcl}y-1& =& 0\\ y& =& 1\\ x+y+5& =& 0\\ x+y& =& -5\end{array}$

Step 2: Solve for x and y

Put the value of y in another equation and solve then;

 $\begin{array}{rcl}x+1& =& -5\\ x& =& -6\end{array}$ 

Thus, the only critical point is (-6, 1).