Fundamentals of Differential Equations and Boundary Value Problems
7th Edition
ISBN: 9780321977106
Author: Nagle, R. Kent
Publisher: Pearson Education, Limited
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Chapter 12.2, Problem 14E
In Problems 13-20, classify the critical point at the origin and, either by hand or using a computer software package, sketch a phase plane diagram for the given system.
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For the following two-population system, first describe the type of x- and y-populations involved (exponential
or logistic) and the nature of their interaction-competition, cooperation, or predation. Then find and characterize
the system's critical points (as to type and stability). Determine what nonzero x- and y-populations can
coexist. Finally, construct a phase plane portrait that enables you to describe the long-term behavior of the two
populations in terms of their initial populations x(0) and y(0).
dx
dt
dy
dt=xy-4y
= 5xy-10x
CICCES
Describe the type of x- and y-populations involved. Select the correct choice below.
OA. The populations involved are naturally declining populations in competition.
OB. The populations involved are naturally growing populations in cooperation.
OC. The populations involved are naturally declining populations in cooperation.
OD. The populations involved are naturally growing populations in competition.
Interaction of two species of squirrels fiercely competing for the same ecological niche on
an island is described by Lotka-Volterra-Gause equations
dN1
N1(2 – N1 – 2N2) = f(N1, N2),
dt
(1)
dN2
N2(3 – N2 – 3N1) = g(N1, N2),
dt
where N1 = N1(t) and N2 = N2(t) are the population densities of the competing species.
3. The steady-state distribution of temperature on a heated plate can
be modeled by the Laplace equation,
25°C
25°C
If the plate is represented by a
series of nodes (Fig.1), centered
T12
100°C
O°C
finite-divided
differences
can
substituted
for
the
second
T
100°C
0°C
derivatives, which results in a
system of linear algebraic equations
as follows:
75°C
75°C
Use the Gauss-Seidel method to
solve for the temperatures of the
(175
|125
75
25
-1 -1
4
-1
4
nodes
in Fig.1.
Perform
the
0 - 1||T,
2
4
-1|T21
-
computation until ɛ, is less than Es
= 0.5%.
-1 -1
4
[T2
MATH206 week (5)
45
Spring 2021, 20/4/2021
Chapter 12 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 12.2 - In Problem 16, classify the critical point at the...Ch. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - Prob. 7ECh. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.2 - In Problem 712, find and classify the critical...
Ch. 12.2 - In Problem 712, find and classify the critical...Ch. 12.2 - In Problem 712, find and classify the critical...Ch. 12.2 - Prob. 13ECh. 12.2 - In Problems 13-20, classify the critical point at...Ch. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - In Problems 13-20, classify the critical point at...Ch. 12.2 - Prob. 19ECh. 12.2 - Prob. 20ECh. 12.2 - Show that when the system x(t)=ax+by+p,...Ch. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - Prob. 25ECh. 12.2 - Show when the roots of the characteristic equation...Ch. 12.2 - Prob. 27ECh. 12.3 - In Problems 1 -8, show that the given system is...Ch. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - Prob. 7ECh. 12.3 - Prob. 8ECh. 12.3 - In Problems 9 -12, find all the critical points...Ch. 12.3 - Prob. 10ECh. 12.3 - Prob. 11ECh. 12.3 - In Problems 9 -12, find all the critical points...Ch. 12.3 - In Problems 13-16, convert the second-order...Ch. 12.3 - In Problems 13-16, convert the second-order...Ch. 12.3 - Prob. 15ECh. 12.3 - Prob. 16ECh. 12.3 - Prob. 17ECh. 12.3 - Prob. 18ECh. 12.3 - Prob. 19ECh. 12.3 - Prob. 20ECh. 12.3 - van der Pols Equation. a. Show that van der Pols...Ch. 12.3 - Consider the system dxdt=(+)x+y, dydt=x+(+)y,...Ch. 12.3 - Prob. 23ECh. 12.3 - Show that coexistence occurs in the competing...Ch. 12.3 - When one of the populations in a competing species...Ch. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Prob. 9ECh. 12.4 - Prob. 10ECh. 12.4 - Prob. 11ECh. 12.4 - Prob. 12ECh. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - Prob. 18ECh. 12.4 - Prob. 19ECh. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - Prob. 4ECh. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - Prob. 6ECh. 12.5 - Prob. 7ECh. 12.5 - Prob. 8ECh. 12.5 - In problem 9-14, use Lyapunovs direct method to...Ch. 12.5 - In problem 9-14, use Lyapunovs direct method to...Ch. 12.5 - Prob. 11ECh. 12.5 - Prob. 12ECh. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - Prob. 15ECh. 12.5 - Prob. 16ECh. 12.5 - Prove that the zero solution for a conservative...Ch. 12.6 - Semistable Limit cycle. For the system...Ch. 12.6 - Prob. 2ECh. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - In Problems 512, either by hand or using a...Ch. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - In Problems 5-12, either by hand or using computer...Ch. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - In Problems 5-12, either by hand or using computer...Ch. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - Prob. 15ECh. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - Prob. 20ECh. 12.6 - Prob. 21ECh. 12.6 - Prob. 22ECh. 12.6 - Prob. 23ECh. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - Prob. 26ECh. 12.6 - Prob. 27ECh. 12.6 - Prob. 28ECh. 12.7 - Prob. 1ECh. 12.7 - Prob. 2ECh. 12.7 - Prob. 3ECh. 12.7 - Prob. 4ECh. 12.7 - Prob. 5ECh. 12.7 - Prob. 6ECh. 12.7 - Prob. 9ECh. 12.7 - Prob. 10ECh. 12.7 - Prob. 11ECh. 12.7 - Prob. 12ECh. 12.7 - Prob. 13ECh. 12.7 - Prob. 14ECh. 12.7 - Prob. 15ECh. 12.7 - Prob. 16ECh. 12.7 - Prob. 17ECh. 12.7 - Prob. 18ECh. 12.8 - Calculate the Jacobian eigenvalues at the critical...Ch. 12.8 - Prob. 2ECh. 12.8 - Prob. 3ECh. 12.8 - Prob. 4ECh. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - Prob. 2RPCh. 12.RP - Prob. 3RPCh. 12.RP - Prob. 4RPCh. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - Prob. 7RPCh. 12.RP - In Problems 7 and 8, use the potential plane to...Ch. 12.RP - In Problems 9-12, use Lyapunovs direct method to...Ch. 12.RP - Prob. 10RPCh. 12.RP - In Problems 9-12, use Lyapunovs direct method to...Ch. 12.RP - Prob. 12RPCh. 12.RP - Prob. 13RPCh. 12.RP - In Problem 13 and 14, sketch the phase plane...Ch. 12.RP - In Problems 15 and 16, determine whether the given...Ch. 12.RP - Prob. 16RPCh. 12.RP - In Problems 17 and 18, determine the stability of...Ch. 12.RP - In Problems 17 and 18, determine the stability of...
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