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Found in: Page 272

### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069

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

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

See the step by step solution

## Step 1: Find the critical point.

Here the equation is:

$\frac{dx}{dt}=y\phantom{\rule{0ex}{0ex}}\frac{dy}{dt}=-x+{x}^{2}$

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.