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 5.8, Problem 5E
To determine
To find:
The points of Poincare map, graph them with the given values for the equation, and describe the attractor for the given system.
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Chapter 5 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
Ch. 5.2 - Let A=D1, B=D+2, C=D2+D2, where D=d/dt. For y=t38,...Ch. 5.2 - Show that the operator (D1)(D+2) is the same as...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...
Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - Prob. 14ECh. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - Prob. 16ECh. 5.2 - Prob. 17ECh. 5.2 - In Problems 3-18, use the elimination method to...Ch. 5.2 - In Problems 19-21, solve the given initial value...Ch. 5.2 - In Problems 19-21, solve the given initial value...Ch. 5.2 - Prob. 21ECh. 5.2 - Prob. 22ECh. 5.2 - Prob. 23ECh. 5.2 - Prob. 24ECh. 5.2 - In Problems 25-28, use the elimination method to...Ch. 5.2 - Prob. 26ECh. 5.2 - Prob. 27ECh. 5.2 - Prob. 28ECh. 5.2 - Prob. 29ECh. 5.2 - Prob. 30ECh. 5.2 - Two large tanks, each holding 100L of liquid, are...Ch. 5.2 - In Problem 31, 3L/min of liquid flowed from tank A...Ch. 5.2 - In Problem 31, assume that no solution flows out...Ch. 5.2 - Feedback System with Pooling Delay. Many physical...Ch. 5.2 - Arms Race. A simplified mathematical model for an...Ch. 5.2 - Let A, B, and C represent three linear...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - In Problems 1-7, convert the given initial value...Ch. 5.3 - Prob. 8ECh. 5.3 - In Section 3.6, we discussed the improved Eulers...Ch. 5.3 - In Problems 10-13, use the vectorized Euler method...Ch. 5.3 - In Problems 10-13, use the vectorized Euler method...Ch. 5.3 - In Problems 10-13, use the vectorized Euler method...Ch. 5.3 - In Problems 10-13, use the vectorized Euler method...Ch. 5.3 - Prob. 14ECh. 5.3 - In Problems 14-24, you will need a computer and a...Ch. 5.3 - In Problems 14-24, you will need a computer and a...Ch. 5.3 - Prob. 18ECh. 5.3 - Prob. 19ECh. 5.3 - In Problems 14-24, you will need a computer and a...Ch. 5.3 - Prob. 21ECh. 5.3 - Prob. 22ECh. 5.3 - Prob. 24ECh. 5.3 - Prob. 25ECh. 5.3 - Prob. 26ECh. 5.3 - Prob. 27ECh. 5.3 - Prob. 28ECh. 5.3 - In Problems 25-30, use a software package or the...Ch. 5.3 - Prob. 30ECh. 5.4 - In Problems 1 and 2, verify that the pair x(t),...Ch. 5.4 - In Problems 1 and 2, verify that pair x(t), y(t)...Ch. 5.4 - In Problems 3-6, find the critical point set for...Ch. 5.4 - Prob. 4ECh. 5.4 - In Problems 3-6, find the critical point set for...Ch. 5.4 - In Problems 3-6, find the critical point set for...Ch. 5.4 - In Problems 7-9, solve the related phase plane...Ch. 5.4 - In Problems 7-9, solve the related phase plane...Ch. 5.4 - In Problems 7-9, solve the related phase plane...Ch. 5.4 - Find all the critical points of the system...Ch. 5.4 - In Problems 11-14, solve the related phase plane...Ch. 5.4 - In Problems 11-14, solve the related phase plane...Ch. 5.4 - In Problems 11-14, solve the related phase plane...Ch. 5.4 - In Problems 11-14, solve the related phase plane...Ch. 5.4 - In Problems 15-18, find all critical points for...Ch. 5.4 - In Problems 15-18, find all critical points for...Ch. 5.4 - In Problems 15-18, find all critical points for...Ch. 5.4 - In Problems 15-18, find all critical points for...Ch. 5.4 - In Problems 19-24, convert the given second-order...Ch. 5.4 - In Problems 19-24, convert the given second-order...Ch. 5.4 - Prob. 21ECh. 5.4 - In Problems 19-24, convert the given second-order...Ch. 5.4 - Prob. 23ECh. 5.4 - In Problems 19-24, convert the given second-order...Ch. 5.4 - Prob. 25ECh. 5.4 - Prob. 26ECh. 5.4 - Prob. 27ECh. 5.4 - Prob. 28ECh. 5.4 - A proof of Theorem 1, page 266, is outlined below....Ch. 5.4 - Phase plane analysis provides a quick derivation...Ch. 5.4 - Prob. 32ECh. 5.4 - Prob. 34ECh. 5.4 - Sticky Friction. An alternative for the damping...Ch. 5.4 - Rigid Body Nutation. Eulers equations describe the...Ch. 5.5 - Radioisotopes and Cancer Detection. A radioisotope...Ch. 5.5 - Secretion of Hormones. The secretion of hormones...Ch. 5.5 - Prove that the critical point (8) of the...Ch. 5.5 - Suppose for a certain disease described by the SIR...Ch. 5.5 - Prob. 6ECh. 5.5 - Prob. 7ECh. 5.5 - Prob. 8ECh. 5.5 - Prob. 9ECh. 5.5 - Prove that the infected population I(t) in the SIR...Ch. 5.6 - Two springs and two masses are attached in a...Ch. 5.6 - Determine the equations of motion for the two...Ch. 5.6 - Four springs with the same spring constant and...Ch. 5.6 - Two springs, two masses, and a dashpot are...Ch. 5.6 - Referring to the coupled mass-spring system...Ch. 5.6 - Prob. 7ECh. 5.6 - A double pendulum swinging in a vertical plane...Ch. 5.6 - Prob. 9ECh. 5.6 - Suppose the coupled mass-spring system of Problem...Ch. 5.7 - An RLC series circuit has a voltage source given...Ch. 5.7 - An RLC series circuit has a voltage source given...Ch. 5.7 - Prob. 3ECh. 5.7 - An LC series circuit has a voltage source given by...Ch. 5.7 - An RLC series circuit has a voltage source given...Ch. 5.7 - Show that when the voltage source in (4) is of the...Ch. 5.7 - Prob. 7ECh. 5.7 - Prob. 8ECh. 5.7 - Prob. 9ECh. 5.7 - Prob. 10ECh. 5.7 - In Problems 10-13, find a system of differential...Ch. 5.7 - In Problems 10-13, find a system of differential...Ch. 5.7 - In Problems 10-13, find a system of differential...Ch. 5.8 - A software package that supports the construction...Ch. 5.8 - Prob. 2ECh. 5.8 - A software package that supports the construction...Ch. 5.8 - Prob. 4ECh. 5.8 - Prob. 5ECh. 5.8 - A software package that supports the construction...Ch. 5.8 - Prob. 11ECh. 5.RP - In Problems 1-4, find a general solution x(t),...Ch. 5.RP - In Problems 1-4, find a general solution x(t),...Ch. 5.RP - In Problems 1-4, find a general solution x(t),...Ch. 5.RP - In Problems 1-4, find a general solution x(t),...Ch. 5.RP - Prob. 5RPCh. 5.RP - Prob. 6RPCh. 5.RP - Prob. 7RPCh. 5.RP - Prob. 8RPCh. 5.RP - Prob. 9RPCh. 5.RP - Prob. 10RPCh. 5.RP - Prob. 11RPCh. 5.RP - Prob. 12RPCh. 5.RP - Prob. 13RPCh. 5.RP - Prob. 14RPCh. 5.RP - Prob. 15RPCh. 5.RP - Prob. 16RPCh. 5.RP - Prob. 17RPCh. 5.RP - In the coupled mass-spring system depicted in...
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- Except for the case of constant solutions, it is usually not possible to find explicit expression for thesolutions of a nonlinear autonomous system, However, if the system contains such an expression as x^2+y^2we can solve some nonlinear systems by changing to polar coordinates.(a) Let x =r cos θ and y = r sin θ where r and θ are functions of t. Express dr/dt and dθ/dt with respect to x, y, dx/dt,dy/dt, and r. b) Consider the following nonlinear plane autonomous system dx/dt = -y-x√x^2+y^2 dy/dt= x-y√x^2+y^2 satisfying the initial condition x(0)=3 and y(0) =3. Express the system (DE1) in polar coordinates. c) Solve the system you rewrote in part (b) for r and θ. d) Sketch the trajectory in part (c). e) Identify and classify the critical point of the system (DE1).arrow_forward14.Consider the vibrating system described by the initial value problem u′′+u=3cos(ωt),u(0)=1,u′(0)=1 a.Find the solution for ω ≠ 1.arrow_forwardQuestion 4: Consider the discrete-time dynamical system Nt+1 = rNt Nt−3 , where r is a parameter. Find all values (if they exist) of parameter r, for which the system has (i) no equilibria, (ii) one equilibrium, (iii) nonnegative equilibriaarrow_forward
- Consider the forced but undamped system described by the initial value problem u′′+u=3cos(ωt),u(0)=0,u′(0)=0.Plot the solution u(t) versus t for ω = 0.7, ω = 0.8, and ω = 0.9. Describe how the response u(t) changes as ω varies in this interval. What happens as ω takes on values closer and closer to 1? Note that the natural frequency of the unforced system is ω0 = 1.arrow_forwardthe consideration of the Rosenzweig–MacArthur predator–prey model introduced in Problem 13 of Section 9.5. Consider the system x′=x(a−0.2x−2yx+6),y′=y(−0.25+xx+6),x′=xa−0.2x−2yx+6,y′=y−0.25+xx+6, where a is a positive parameter. Observe that this system includes the one in Problem 18 above and the one in Problem 13 in Section 9.5. a.Find all of the critical points. b.Consider the critical point in the interior of the first quadrant. Find the eigenvalues of the approximate linear system. Determine the value a0 where this critical point changes from asymptotically stable to unstable. c.Draw a phase portrait for a value of a slightly greater than a0. Observe that a limit cycle has appeared. How does the limit cycle change as a increases further?arrow_forward
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