Differential Equations: An Introduction to Modern Methods and Applications
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 7.2, Problem 27P

A generalization of the damped pendulum equation discussed in the text, or a damped spring-mass system, is the Lienard equation

d 2 x d t 2 + c ( x ) d x d t + g ( x ) = 0

If c ( x ) is a constant and g ( x ) = k x , then this equation has the form of the linear pendulum equation [replace sin θ with θ in Eq. ( 13 ) of Section 7.11 ; otherwise, the damping force

c ( x ) d x / d t and the restoring force g ( x ) are nonlinear. Assume that c is continuously differentiable, g is twice continuously differentiable, and g ( 0 ) = 0 .

(a) Write the Lienard equation as a system of two first order equations by introducing the variable y = d x / d t .

(b) Show that ( 0 , 0 ) is a critical point and that the system is almost linear in the neighbourhood of ( 0 , 0 ) .

(c) Show that if c ( 0 ) > 0 and g ' ( 0 ) > 0 , then the critical point is asymptotically stable, and that if c ( 0 ) < 0 or g 1 ( 0 ) < 0 , then the critical point is unstable.

Hint: Use Taylor series to approximate c and g in the neighbourhood of x=0 .

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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 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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