Explanation Given: The system is, d x 1 d t = x 1 ( 1 − x 1 − 1 2 x 2 − x 3 ) d x 2 d t = x 2 ( 1 − x 1 − x 2 − 1 2 x 3 ) d x 3 d t = x 3 ( 1 − 1 2 x 1 − x 2 − x 3 ) Approach: Critical points can be obtained by equating the equations to zero and solve for the variables. The characteristic equation is given by, | A − λ I | = 0 Here, I is the identity matrix of the same dimension as matrix A . Then find the eigenvalues of the given matrix from the characteristic equation and check the stability of the corresponding equilibrium solutions. Calculation: The critical points are obtained by, Equate the above equations to zero. x 1 ( 1 − x 1 − 1 2 x 2 − x 3 ) = 0 … ( 1 ) x 1 = 0 x 2 ( 1 − x 1 − x 2 − 1 2 x 3 ) = 0 … ( 2 ) x 2 = 0 x 3 ( 1 − 1 2 x 1 − x 2 − x 3 ) = 0 x 3 = 0 … ( 3 ) Thus, one of the critical points is ( 0 , 0 , 0 ) . Now, put x 1 = 0 , x 2 = 0 . From equation ( 3 ) , x 3 = 1 Put x 1 = 0 , x 2 = 1 . From equation ( 2 ) , x 3 = 0 Similarly, put x 1 = 1 , x 2 = 0 . From equation ( 1 ) , x 3 = 0 Other three critical points obtained are ( 0 , 0 , 1 ) , ( 0 , 1 , 0 ) , and ( 1 , 0 , 0 ) . Another critical point is obtained at the intersection of equations ( 1 ) , ( 2 ) , ( 3 ) . Now, Equation ( 1 ) can be written as, x 1 + 1 2 x 2 + x 3 = 1 … ( 4 ) Similarly, equation ( 2 ) can be written as, x 1 + x 2 + 1 2 x 3 = 1 … ( 5 ) And, equation ( 3 ) can be written as, 1 2 x 1 + x 2 + x 3 = 1 … ( 6 ) Subtract equation ( 5 ) from equation ( 4 ) . x 1 + 1 2 x 2 + x 3 − ( x 1 + x 2 + 1 2 x 3 ) = 1 − 1 − 1 2 x 2 + 1 2 x 3 = 0 x 2 = x 3 Substitute x 3 for x 2 in equations ( 5 ) and ( 6 ) and then solve them. x 1 + 3 2 x 3 = 1 1 2 x 1 + 2 x 3 = 1 Multiply equation 1 2 x 1 + 2 x 3 = 1 by 2 and subtract from equation x 1 + 3 2 x 3 = 1 , the following is obtained

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 12.7, Problem 17E

To determine

To find:

The critical points for the system and the stability of the corresponding equilibrium solutions.

dx1dt=x1(1−x1−12x2−x3)

dx2dt=x2(1−x1−x2−12x3)

dx3dt=x3(1−12x1−x2−x3)

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Chapter 12.7, Problem 16E

Chapter 12.7, Problem 18E

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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. 2E Ch. 12.2 - Prob. 3E Ch. 12.2 - Prob. 4E Ch. 12.2 - Prob. 5E Ch. 12.2 - Prob. 6E Ch. 12.2 - Prob. 7E Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E 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 - In Problem 712, find and classify the critical...Ch. 12.2 - Prob. 13E Ch. 12.2 - In Problems 13-20, classify the critical point at...Ch. 12.2 - Prob. 15E Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - In Problems 13-20, classify the critical point at...Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Show that when the system x(t)=ax+by+p,...Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Prob. 24E Ch. 12.2 - Prob. 25E Ch. 12.2 - Show when the roots of the characteristic equation...Ch. 12.2 - Prob. 27E Ch. 12.3 - In Problems 1 -8, show that the given system is...Ch. 12.3 - Prob. 2E Ch. 12.3 - Prob. 3E Ch. 12.3 - Prob. 4E Ch. 12.3 - Prob. 5E Ch. 12.3 - Prob. 6E Ch. 12.3 - Prob. 7E Ch. 12.3 - Prob. 8E Ch. 12.3 - In Problems 9 -12, find all the critical points...Ch. 12.3 - Prob. 10E Ch. 12.3 - Prob. 11E Ch. 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. 15E Ch. 12.3 - Prob. 16E Ch. 12.3 - Prob. 17E Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - Prob. 20E Ch. 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. 23E Ch. 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. 1E Ch. 12.4 - Prob. 2E Ch. 12.4 - Prob. 3E Ch. 12.4 - Prob. 4E Ch. 12.4 - Prob. 5E Ch. 12.4 - Prob. 6E Ch. 12.4 - Prob. 7E Ch. 12.4 - Prob. 8E Ch. 12.4 - Prob. 9E Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Prob. 12E Ch. 12.4 - Prob. 13E Ch. 12.4 - Prob. 14E Ch. 12.4 - Prob. 15E Ch. 12.4 - Prob. 16E Ch. 12.4 - Prob. 17E Ch. 12.4 - Prob. 18E Ch. 12.4 - Prob. 19E Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E 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 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - Prob. 4E Ch. 12.5 - In Problems 1-8, use Lyapunovs direct method to...Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 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. 11E Ch. 12.5 - Prob. 12E Ch. 12.5 - Prob. 13E Ch. 12.5 - Prob. 14E Ch. 12.5 - Prob. 15E Ch. 12.5 - Prob. 16E Ch. 12.5 - Prove that the zero solution for a conservative...Ch. 12.6 - Semistable Limit cycle. For the system...Ch. 12.6 - Prob. 2E Ch. 12.6 - Prob. 3E Ch. 12.6 - Prob. 4E Ch. 12.6 - In Problems 512, either by hand or using a...Ch. 12.6 - Prob. 6E Ch. 12.6 - Prob. 7E Ch. 12.6 - Prob. 8E Ch. 12.6 - In Problems 5-12, either by hand or using computer...Ch. 12.6 - Prob. 10E Ch. 12.6 - Prob. 11E Ch. 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. 15E Ch. 12.6 - In Problems 13-18, show that the given system or...Ch. 12.6 - Prob. 17E Ch. 12.6 - Prob. 18E Ch. 12.6 - Prob. 19E Ch. 12.6 - Prob. 20E Ch. 12.6 - Prob. 21E Ch. 12.6 - Prob. 22E Ch. 12.6 - Prob. 23E Ch. 12.6 - Prob. 24E Ch. 12.6 - Prob. 25E Ch. 12.6 - Prob. 26E Ch. 12.6 - Prob. 27E Ch. 12.6 - Prob. 28E Ch. 12.7 - Prob. 1E Ch. 12.7 - Prob. 2E Ch. 12.7 - Prob. 3E Ch. 12.7 - Prob. 4E Ch. 12.7 - Prob. 5E Ch. 12.7 - Prob. 6E Ch. 12.7 - Prob. 9E Ch. 12.7 - Prob. 10E Ch. 12.7 - Prob. 11E Ch. 12.7 - Prob. 12E Ch. 12.7 - Prob. 13E Ch. 12.7 - Prob. 14E Ch. 12.7 - Prob. 15E Ch. 12.7 - Prob. 16E Ch. 12.7 - Prob. 17E Ch. 12.7 - Prob. 18E Ch. 12.8 - Calculate the Jacobian eigenvalues at the critical...Ch. 12.8 - Prob. 2E Ch. 12.8 - Prob. 3E Ch. 12.8 - Prob. 4E Ch. 12.RP - In Problems 1-6, find all the critical points for...Ch. 12.RP - Prob. 2RP Ch. 12.RP - Prob. 3RP Ch. 12.RP - Prob. 4RP Ch. 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. 7RP Ch. 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. 10RP Ch. 12.RP - In Problems 9-12, use Lyapunovs direct method to...Ch. 12.RP - Prob. 12RP Ch. 12.RP - Prob. 13RP Ch. 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. 16RP Ch. 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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The critical points for the system and the stability of the corresponding equilibrium solutions. d x 1 d t = x 1 ( 1 − x 1 − 1 2 x 2 − x 3 ) d x 2 d t = x 2 ( 1 − x 1 − x 2 − 1 2 x 3 ) d x 3 d t = x 3 ( 1 − 1 2 x 1 − x 2 − x 3 )

Solution Summary: The author explains the critical points for the system and the stability of the corresponding equilibrium solutions.

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To find:

The critical points for the system and the stability of the corresponding equilibrium solutions.

dx1dt=x1(1−x1−12x2−x3)

dx2dt=x2(1−x1−x2−12x3)

dx3dt=x3(1−12x1−x2−x3)

Want to see the full answer?

Chapter 12 Solutions

The critical points for the system and the stability of the corresponding equilibrium solutions. d x 1 d t = x 1 ( 1 − x 1 − 1 2 x 2 − x 3 ) d x 2 d t = x 2 ( 1 − x 1 − x 2 − 1 2 x 3 ) d x 3 d t = x 3 ( 1 − 1 2 x 1 − x 2 − x 3 )

Solution Summary: The author explains the critical points for the system and the stability of the corresponding equilibrium solutions.

Concept explainers

Videos

To find: The critical points for the system and the stability of the corresponding equilibrium solutions. dx1dt=x1(1−x1−12x2−x3) dx2dt=x2(1−x1−x2−12x3) dx3dt=x3(1−12x1−x2−x3)

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Chapter 12 Solutions

To find:

The critical points for the system and the stability of the corresponding equilibrium solutions.

dx1dt=x1(1−x1−12x2−x3)

dx2dt=x2(1−x1−x2−12x3)

dx3dt=x3(1−12x1−x2−x3)