Q28E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Figure 5.16 displays some trajectories for the system $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{y}\mathbf{,}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathit{x}\mathbf{+}{\mathit{x}}^{\mathbf{2}}$What types of critical points (compare Figure 5.12 on page 267) occur at (0, 0) and (1, 0)?

Step-by-Step Solution

Answer

The points are (0,0) and saddle point(1,0).

Step 1: Find the critical point.

Here the equation is:

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

 

And 

$\frac{dy}{dx}=\frac{-x+x^2}{y}$

 

Here the points are (0,0) and saddle point (1,0).

Step 2: Sketch Directional field.

This is the required result.