Q3E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

$\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{=}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{,}\frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{1}$

Step-by-Step Solution

Answer

The critical points are $\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right),\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$

Step 1: Find critical points

Consider the system as:

 $$\begin{gathered} \frac{dx}{dt}=x-y \\ \frac{dy}{dt}=x^2+y^2-1 \end{gathered}$$

 

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

So, 

$\begin{array}{rcl}x-y& =& 0\\ x& =& y\\ x^2+y^2-1& =& 0\\ x^2+y^2& =& 1\end{array}$

Step 2: Solve for x and y

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

 $\begin{array}{rcl}{x}^2+x^2& =& 1\\ 2x^2& =& 1\\ x& =& \pm \frac{1}{\sqrt{2}}\end{array}$

Thus, the critical points are $\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)\mathbf{,}\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$.

This is the required result.