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Expert-verified Found in: Page 271 ### 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 # In Problems 15–18, find all critical points for the given system. Then use a software package to sketch the direction field in the phase plane and from this description the stability of the critical points (i.e., compare with Figure 5.12).$\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbf{2}}{\mathbit{x}}{\mathbf{+}}{\mathbf{13}}{\mathbit{y}}{\mathbf{,}}\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{t}}{\mathbf{=}}{\mathbf{-}}{\mathbit{x}}{\mathbf{-}}{\mathbf{2}}{\mathbit{y}}$

This is a stable node point is (0,0).

See the step by step solution

## Step 1: Find critical points

Here the system is;

$\frac{dx}{dt}=2x+13y\phantom{\rule{0ex}{0ex}}\frac{dy}{dt}=-x-2y\phantom{\rule{0ex}{0ex}}$

For critical points equate the system equal to zero.

$2x+13y=0\phantom{\rule{0ex}{0ex}}-x-2y=0\phantom{\rule{0ex}{0ex}}$

Solve for x and y by eliminating the method.

The values of x=0 and y=0.

So, this is the stable node point (0,0).

## Step 2: Sketch This is the required result.

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