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Fundamentals Of Differential Equations And Boundary Value Problems
Found in: Page 271
Fundamentals Of Differential Equations And Boundary Value Problems

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

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Short Answer

Find all the critical points of the system

dxdt=x2-1dydt=xy

and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

The result is

for y>0,x>1,y=ecx2-1for y>0,x<1,y=ec1-x2for y<0,x>1,y=-ecx2-1for y<0,x<1,y=-ec1-x2

And the end points are (-1,0) (1,0).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find critical points

For a critical point put the system equal to 0.

x2-1=0xy=0x2=1x=±1y=0

Step 2: Find the value of y

Now,

dydx=xyx2-1∫1ydy=∫x2x2-1lny=12lnt+c             bysubtitutionlny=lnt12+cy=ect12y=ecx2-1

Step 3: Prove results that endpoints are semicircles.

Now there are two cases when y > 0 and y < 0 for both cases x2-1<0  and  x2-1>0 then x<1  and  x>1

for y>0,x>1,y=ecx2-1fory>0,x<1,y=ec1-x2fory<0,x>1,y=-ecx2-1fory<0,x<1,y=-ec1-x2

The trajectories that possibly lie on semicircles. If we square the equation then

y2=e2c(1-x2)

This will be the equation of circle only if e2c=1. Therefore, only two solutions lie on the semicircle, y=x2-1,y=-x2-1.

Therefore, it is a circle of radius 1 center at origin the endpoints are-1,0,  1,0.

Most popular questions for Math Textbooks

Rigid Body Nutation. Euler’s equations describe the motion of the principal-axis components of the angular velocity of a freely rotating rigid body (such as a space station), as seen by an observer rotating with the body (the astronauts, for example). This motion is called nutation. If the angular velocity components are denoted by x, y, and z, then an example of Euler’s equations is the three-dimensional autonomous system

\(\begin{array}{l}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = yz}}\\\frac{{{\bf{dy}}}}{{{\bf{dt}}}}{\bf{ = - 2xz}}\\\frac{{{\bf{dz}}}}{{{\bf{dt}}}}{\bf{ = xy}}\end{array}\)

The trajectory of a solution x(t),y(t), z(t) to these equations is the curve generated by the points (x(t), y(t), z(t) ) in xyz-phase space as t varies over an interval I.

(a) Show that each trajectory of this system lies on the surface of a (possibly degenerate) sphere centered at the origin (0, 0, 0).[Hint: Compute\(\frac{{\bf{d}}}{{{\bf{dt}}}}{\bf{(}}{{\bf{x}}^{\bf{2}}}{\bf{ + }}{{\bf{y}}^{\bf{2}}}{\bf{ + }}{{\bf{z}}^{\bf{2}}}{\bf{)}}\)What does this say about the magnitude of the angular velocity vector?

(b) Find all the critical points of the system, i.e., all points\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{y}}_{\bf{o}}}{\bf{,}}{{\bf{z}}_{\bf{o}}}{\bf{)}}\) such that \({\bf{x(t) = }}{{\bf{x}}_{\bf{o}}}{\bf{,y(t) = }}{{\bf{y}}_{\bf{o}}}{\bf{,z(t) = }}{{\bf{z}}_{\bf{o}}}\) is a solution. For such solutions, the angular velocity vector remains constant in the body system.

(c) Show that the trajectories of the system lie along the intersection of a sphere and an elliptic cylinder of the form\({{\bf{y}}^{\bf{2}}}{\bf{ + 2}}{{\bf{x}}^{\bf{2}}}{\bf{ = C}}\) for some constant C. [Hint: Consider the expression for dy/dx implied by Euler’s equations.]

(d) Using the results of parts (b) and (c), argue that the trajectories of this system are closed curves. What does this say about the corresponding solutions?

(e) Figure 5.19 displays some typical trajectories for this system. Discuss the stability of the three critical points indicated on the positive axes.

In Problem 36, if a small furnace that generates 1000 Btu/hr is placed in zone B, determine the coldest it would eventually get in zone B has a heat capacity of 2°F per thousand Btu.

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.

D-3x+D-1y=t,D+1x+D+4y=1

Show that the operator (D-1)(D+2) is the same as the operator D2+D-2.

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).

dxdt=2x+13y,dydt=-x-2y
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