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Q17E

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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 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).
$\frac{d x}{d t} = 2 x + 13 y, \frac{d y}{d t} = - x - 2 y$

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

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Find critical points

Here the system is;

$\frac{d x}{d t} = 2 x + 13 y \frac{d y}{d t} = - x - 2 y$

For critical points equate the system equal to zero.

$2 x + 13 y = 0 - x - 2 y = 0$

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.

Most popular questions for Math Textbooks

Fluid Ejection. In the design of a sewage treatment plant, the following equation arises: $60 - H = (77 . 7) H'' + (19 . 42) (H')^{2}; H (0) = H' (0) = 0$ where H is the level of the fluid in an ejection chamber, and t is the time in seconds. Use the vectorized Runge–Kutta algorithm with h = 0.5 to approximate $H (t)$ over the interval [0, 5].

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).

$\frac{d^{2} x}{d t^{2}} - y = 0$

Suppose the displacement functions $x (t)$ and $y (t)$ for a coupled mass-spring system (similar to the one discussed in Problem 6) satisfy the initial value problem

$\begin{array}{rcl} x'' (t) + 5 x (t) - 2 y (t) = 0 \\ y'' (t) + 2 y (t) - 2 x (t) = 3 \sin 2 t \\ x (0) = x' (0) = 0 \\ y (0) = 1, y' (0) = 0 \end{array}$

Solve for $x (t)$ and $y (t)$

Sticky Friction. An alternative for the damping friction model F = -by′ discussed in Section 4.1 is the “sticky friction” model. For a mass sliding on a surface as depicted in Figure 5.18, the contact friction is more complicated than simply -by′. We observe, for example, that even if the mass is displaced slightly off the equilibrium location y = 0, it may nonetheless remain stationary due to the fact that the spring force -ky is insufficient to break the static friction’s grip. If the maximum force that the friction can exert is denoted by m, then a feasible model is given by

\({{\bf{F}}_{{\bf{friction}}}}{\bf{ = }}\left\{ \begin{array}{l}{\bf{ky,if}}\left| {{\bf{ky}}} \right|{\bf{ < }}\mu {\bf{andy' = 0}}\\\mu {\bf{sign(y),if}}\left| {{\bf{ky}}} \right| \ge {\bf{0andy' = 0}}\\ - \mu {\bf{sign(y'),ify'}} \ne 0.\end{array} \right.\)

(The function sign (s) is +1 when s 7 0, -1 when s 6 0, and 0 when s = 0.) The motion is governed by the equation (16) \({\bf{m}}\frac{{{{\bf{d}}^{\bf{2}}}{\bf{y}}}}{{{\bf{d}}{{\bf{t}}^{\bf{2}}}}}{\bf{ = - ky + }}{{\bf{F}}_{{\bf{friction}}}}\)Thus, if the mass is at rest, friction balances the spring force if \(\left| {\bf{y}} \right|{\bf{ < }}\frac{\mu }{{\bf{k}}}\)but simply opposes it with intensity\(\mu \)if\(\left| {\bf{y}} \right| \ge \frac{\mu }{{\bf{k}}}\). If the mass is moving, friction opposes the velocity with the same intensity\(\mu \).

Taking m =\(\mu \) = k = 1, convert (16) into the firstorder system y′ = v (17)\({\bf{v' = }}\left\{ \begin{array}{l}{\bf{0,if}}\left| {\bf{y}} \right|{\bf{ < 1andv = 0}}{\bf{.}}\\{\bf{ - y + sign(y),if}}\left| {\bf{y}} \right| \ge {\bf{1andv = 0}}\\{\bf{ - y - sign(v),ifv}} \ne 0\end{array} \right.\) ,
Form the phase plane equation for (17) when v ≠ 0 and solve it to derive the solutions\({{\bf{v}}^{\bf{2}}}{\bf{ + (y \pm 1}}{{\bf{)}}^{\bf{2}}}{\bf{ = c}}\).where the plus sign prevails for v>0 and the minus sign for v<0.
Identify the trajectories in the phase plane as two families of concentric semicircles. What is the centre of the semicircles in the upper half-plane? The lower half-plane?
What are the critical points for (17)?
Sketch the trajectory in the phase plane of the mass released from rest at y = 7.5. At what value for y does the mass come to rest?

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.

$2 x' + y' - x - y = e^{- t}, x' + y' + 2 x + y = e^{t}$

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