Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Chapter 7.P1, Problem 7P
Use the equation
Then use the qualitative methods of Chapter 7 and numerical simulations to discover as much as you can about the properties of the system (3). Compare and contrast your finding with the properties of solutions of the SIR model.
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stacie is a resident at a medical facility you work at. You are asked to chart the amount of solid food that she consumes.For the noon meal today, she ate 1/2 of a 3 ounce serving of meatloaf, 3/4 of her 3 ounce serving of mashed potatoes, and 1/3 of her 2 ounce serving of green beans. Show in decimal form how many ounces of solid food that Stacie consumed
Find the area of the shaded region.
(a)
5-
y
3
2-
(1,4)
(5,0)
1
3
4
5
6
(b)
3 y
2
Decide whether the problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to
estimate the solution.
STEP 1: Consider the figure in part (a). Since this region is simply a triangle, you may use precalculus methods to solve this part of the problem. First determine the height of the triangle and the length of the triangle's base.
height 4
units
units
base
5
STEP 2: Compute the area of the triangle by employing a formula from precalculus, thus finding the area of the shaded region in part (a).
10
square units
STEP 3: Consider the figure in part (b). Since this region is defined by a complicated curve, the problem seems to require calculus. Find an approximation of the shaded region by using a graphical approach. (Hint: Treat the shaded regi
as…
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Chapter 7 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...
Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problems 1 through 18:...Ch. 7.1 - For each of the systems in Problemsthrough:
Find...Ch. 7.1 -
Consider the equations of motion of an undamped...Ch. 7.1 - The motion of a certain undamped pendulum is...Ch. 7.1 - Consider the pendulum equations dxdt=y,dydt=6sinx....Ch. 7.1 - Prob. 22PCh. 7.1 - Given that x=(t),y=(t) is a solution of the...Ch. 7.1 - Prove that, for the system...Ch. 7.1 - Prove that if a trajectory starts at a noncritical...Ch. 7.1 - Assuming that the trajectory corresponding to a...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems 1 through 20: (a) Determine...Ch. 7.2 - In each of Problems through
Determine all...Ch. 7.2 - Consider the autonomous system dxdt=y,dydt=x+2x3....Ch. 7.2 - Consider the autonomous system
...Ch. 7.2 - The equations of motion of a certain nonlinear...Ch. 7.2 - Theorem 7.2.2 provides no information about the...Ch. 7.2 - In this problem, we show how small changes in the...Ch. 7.2 - In this problem, we show how small changes in the...Ch. 7.2 - A generalization of the damped pendulum equation...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Each of Problems 1 through 6 can be interpreted as...Ch. 7.3 - Show that (1X+2Y)24(1212)XY=(1X2Y)2+412XY. Hence...Ch. 7.3 - Consider the system (2) in the text, and assume...Ch. 7.3 - Consider the system (3) in Example 1 of the text....Ch. 7.3 - The system x=yy=yx(x0.15)(x3) Results from an...Ch. 7.3 - Bifurcation points. Consider the system...Ch. 7.3 - Bifurcation points. Consider the system
Where is...Ch. 7.3 - Bifurcation points. Consider the system
Where is...Ch. 7.3 - Bifurcation points. Consider the system
Where is...Ch. 7.3 - In each of Problem 15 and 16: a) Find the critical...Ch. 7.3 - In each of Problem 15 and 16:
Find the critical...Ch. 7.3 - Suppose that a certain pair of competing species...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - Each of Problems 1 through 5 can be interpreted as...Ch. 7.4 - In this problem, we examine the phase difference...Ch. 7.4 - a) Find the ratio of the amplitudes of the...Ch. 7.4 -
Find the period of the oscillations of the prey...Ch. 7.4 - Consider the system
Where and are positive...Ch. 7.4 - The average size of the prey and predator...Ch. 7.4 - In Problems 11 and 12, we consider the effect of...Ch. 7.4 - In Problems 11 and 12, we consider the effect of...Ch. 7.4 - In the Lotka-Volterra equations, the interaction...Ch. 7.4 - Harvesting in a Predator-Prey Relationship. In a...Ch. 7.4 - Harvesting in a Predator-Prey Relationship. In a...Ch. 7.4 - Harvesting in a Predator-Prey Relationship. In a...Ch. 7.5 - In each of Problems through , an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - In each of Problems through , an autonomous...Ch. 7.5 - In each of Problems 1 through 6, an autonomous...Ch. 7.5 - If x=rcos,y=rsin, show that...Ch. 7.5 - (a) Show that the system has periodic solutions...Ch. 7.5 - Determine the periodic solutions, if any, of the...Ch. 7.5 - Using Theorem, show that the linear autonomous...Ch. 7.5 - In each of Problems 11 and 12, show that the given...Ch. 7.5 - In each of Problems and , show that the given...Ch. 7.5 - Prob. 13PCh. 7.5 -
By examining the graphs of vs. in Figures , , ...Ch. 7.5 - The equation u(113u2)u+u=0 Is often called the...Ch. 7.5 - Consider the system of equations...Ch. 7.5 - Consider the van der Pol system x=y,y=x+(1x2)y,...Ch. 7.5 - Problems 18 and 19 extend the consideration of the...Ch. 7.5 - Problems 18 and 19 extend the consideration of the...Ch. 7.5 - There are certain chemical reactions in which the...Ch. 7.5 - The system
Is a special case of the...Ch. 7.6 - Problems through ask you to fill in some of the...Ch. 7.6 - Problems through ask you to fill in some of the...Ch. 7.6 -
Ch. 7.6 - Consider the ellipsoid
.
Calculate
along...Ch. 7.6 - In each of Problems 5 through 7, carry out the...Ch. 7.6 - In each of Problems 5 through 7, carry out the...Ch. 7.6 - In each of Problems 5 through 7, carry out the...Ch. 7.6 - For certain intervals, or windows, the Lorenz...Ch. 7.6 - Now consider values of r slightly larger than...Ch. 7.P1 - Assume that , that is, the total size of the...Ch. 7.P1 - The triangular region in the SI-plane is depicted...Ch. 7.P1 - If epidemics are identified with solution...Ch. 7.P1 - Find an equation of the form satisfied by the...Ch. 7.P1 - In the SIR system (1), describe qualitatively the...Ch. 7.P1 - Vaccinated individual are protected from acquiring...Ch. 7.P1 - Use the equation to reduce the SIRS model (3) to...Ch. 7.P2 - Consider again the system
(i)
Which...Ch. 7.P2 - Consider the system dxdt=x(1xy),dydt=y(0.80.6yx),...Ch. 7.P2 - Consider the system (i) in Problem 1, and assume...Ch. 7.P2 - Aconstant-yield model, applied to species x,...Ch. 7.P3 - a) Show that there are no critical points when...Ch. 7.P3 - a) Let c=1.3. Find the critical points and the...Ch. 7.P3 - The limit cycle found in Problem 2 comes into...Ch. 7.P3 -
Let. Find the critical points and the...Ch. 7.P3 -
Let. Find the critical points and the...
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