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Q28E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 272

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

Figure 5.16 displays some trajectories for the system $\frac{d x}{d t} = y, \frac{d y}{d t} = - x + x^{2}$ 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 by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find the critical point.

Here the equation is:

$\frac{d x}{d t} = y \frac{d y}{d t} = - x + x^{2}$

And

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

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

Step 2: Sketch Directional field.

This is the required result.

Most popular questions for Math Textbooks

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).

$\frac{d x}{d t} = \frac{3}{y} \frac{d y}{d t} = \frac{2}{x}$

The motion of a pair of identical pendulums coupled with a spring is modeled by the system

${mx}_{1}^{}'' = - \frac{mg}{l} x_{1} - k (x_{1} - x_{2}) {, mx}_{2}^{}'' = - \frac{mg}{l} x_{2} + k (x_{1} - x_{2})$

for small displacements (see Figure 5.36). Determine the two normal frequencies for the system.

Generalized Blasius Equation. H. Blasius, in his study of the laminar flow of a fluid, encountered an equation of the form $y''' + yy'' = (y')^{2} - 1$ . Use the Runge–Kutta algorithm for systems with h = 0.1 to approximate the solution that satisfies the initial conditions $y (0) = 0, y' (0) = 0, y'' (0) = 1 . 32824$ . Sketch this solution on the interval [0, 2].

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.

$x' = 3 x - 2 y + sint, y' = 4 x - y - cost$

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).

$y'' (t) + y (t) - y (t)^{4} = 0$

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