Q25E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Using the software, sketch the direction field in the phase-plane for the system $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{y}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{x}\mathbf{+}{\mathit{x}}^{\mathbf{3}}$From the sketch, conjecture whether the solution passing through each given point is periodic:

1. (0.25, 0.25)
2. (2, 2)
3. (1, 0)

Step-by-Step Solution

Answer

1. The solution is periodic.
2. The solution is non-periodic.
3. The solution of the critical point (1,0) and periodic.

Step 1: Find the critical point

Here the equation is:

$$\begin{gathered} \frac{dx}{dt}=y \\ \frac{dy}{dt}=-x+x^3 \end{gathered}$$

 

For critical points equate the system equal to zero.

 $\begin{array}{rcl}y& =& 0\\ -x+x^3& =& 0\\ x(-1+x^2)& =& 0\\ x& =& 0 or (-1+x^2)=0\\ x& =& 0,1\end{array}$

Step 2: Sketch

Step 3: Solution of a

The solution passing through the point (0.25,0.25) flows around (0,0) and this is periodic.

Step 4: Result of part (b)

The solution passes through the point (2,2), $y\left(t\right)\to \infty as t\to \infty$and this is not periodic.

Step 5: Find the solution (c)

The solution passing through the point (1,0) is a constant solution and this is periodic.

This is the required result.