Explanation Given: The systems of differential equations: d x d t = a x + b y , d y d t = c x + d y is such that a d − b c = 0 . Approach: Critical points are the solutions of x ′ = 0 and y ′ = 0 . Calculation: Solve x ′ ( t ) = 0 and y ′ ( t ) = 0 . a x + b y = 0 Let a ≠ 0 or b ≠ 0 . Without loss of generality, assume a ≠ 0 and b = 0 . From Equation ( 1 ) , x = 0

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

7th Edition

ISBN: 9780321977182

Author: Nagle, R. Kent, Saff, Edward B., Snider, Arthur David

Publisher: PEARSON

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Chapter 12.2, Problem 22E

To determine

To show:

If ad−bc=0, then there exists critical points system given by

x′(t)=ax+by,y′(t)=cx+dy

which are arbitrarily close to (0,0).

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Chapter 12.2, Problem 21E

Chapter 12.2, Problem 23E

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Chapter 12 Solutions

Fundamentals Of Differential Equations And Boundary Value Problems, Books A La Carte Edition (7th Edition)

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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If a d − b c = 0 , then there exists critical points system given by x ′ ( t ) = a x + b y , y ′ ( t ) = c x + d y which are arbitrarily close to ( 0 , 0 ) .

Solution Summary: The author explains that if ad-bc=0, then there exists critical points system.

Concept explainers

Videos

To show:

If ad−bc=0, then there exists critical points system given by

x′(t)=ax+by,y′(t)=cx+dy

which are arbitrarily close to (0,0).

Want to see the full answer?

Chapter 12 Solutions

If a d − b c = 0 , then there exists critical points system given by x ′ ( t ) = a x + b y , y ′ ( t ) = c x + d y which are arbitrarily close to ( 0 , 0 ) .

Solution Summary: The author explains that if ad-bc=0, then there exists critical points system.

Concept explainers

Videos

To show: If ad−bc=0, then there exists critical points system given by x′(t)=ax+by,y′(t)=cx+dy which are arbitrarily close to (0,0).

Want to see the full answer?

Chapter 12 Solutions

To show:

If ad−bc=0, then there exists critical points system given by

x′(t)=ax+by,y′(t)=cx+dy

which are arbitrarily close to (0,0).