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Q20E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 271

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In Problems 19–24, convert the given second-order equation into a first-order system by setting v=y’. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).
$\frac{d^{2} y}{d t^{2}} + y = 0$

The point is an unstable saddle point (0, 0).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the critical point

Here the equation is $\frac{d^{2} y}{d t^{2}} + y = 0$ .

Put $v = y' a nd v' = y''$

Then the given system can be written as:

$y'' = - y v' = - y$

For critical points equate the system equal to zero.

$v = 0 - y = 0 y = 0$

So, the critical point is (0, 0).

The phase plane equation is:

$\frac{d v}{d y} = \frac{- y}{v} \int v d v = \int - y d y v^{2} + y^{2} = c$

Step 2: Sketch

This is the required result.

Most popular questions for Math Textbooks

In Problems 29 and 30, determine the range of values (if any) of the parameter that will ensure all solutions x(t), and y(t) of the given system remain bounded as $t \to + \infty$ .

$\frac{dx}{dt} = - x + λy, \frac{dy}{dt} = x - y$

In Problems 3 – 18, use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

$\frac{dx}{dt} + x + \frac{dy}{dt} = e^{4 t}, 2 x + \frac{d^{2} y}{{dt}^{2}} = 0$

In Problems 7–9, solve the related phase plane differential equation (2), page 263, for the given system.

$\frac{d x}{d t} = y - 1, \frac{d y}{d t} = e^{x + y}$

In Problems 11–14, solve the related phase plane differential equation for the given system. Then sketch by hand several representative trajectories (with their flow arrows).

$\frac{dx}{dt} = (y - x) (y - 1) \frac{dy}{dt} = (x - y) (x - 1)$

Find the critical points and solve the related phase plane differential equation for the system $\frac{dx}{dt} = (x - 1) (y - 1), \frac{dy}{dt} = y (y - 1)$ .Describe (without using computer software) the asymptotic behavior of trajectories (as role="math" localid="1664302603010" $t \to \infty$ ) that start at

(3, 2)
(2, 1/2)
( -2, 1/2)
(3, -2)

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