Explanation Given: Given differential equation is d x d t = − 3 x − y − 4 , d y d t = 29 x + y + 30 . Approach: Assume the origin ( 0 , 0 ) is the only critical point for the linear system x ′ ( t ) = a x + b y and y ′ ( t ) = c x + d y where a , b , c and d are real number. Let r 1 and r 2 be the Eigen values of the coefficient matrix or equivalently, the root of the characteristic equation r 2 − ( a + d ) r + ( a d − b c ) = 0 . The stability of origin and its classification of origin as a critical point depends on the roots r 1 and r 2 as follows; Roots (eigenvalues) Type of critical points Stability Distinct, positive Improper node Unstable Distinct, negative Improper node Asymptotically stable Opposite sign Saddle point Unstable Equal, positive Proper or improper node Unstable Equal, negative Proper or improper node Asymptotically stable Positive real part of complex value Spiral point Unstable Negative real part of complex value Spiral point Asymptotically stable Pure imaginary center Stable Calculation: Consider the system of linear equation in such a way that d x d t = 0 and d y d t = 0 . − 3 x − y − 4 = 0 − 3 x − y = 4 … ( 1 ) 29 x + y + 30 = 0 29 x + y = − 30 … ( 2 ) Add equation (1) to equation (2). − 3 x − y = 4 29 x + y = − 30 _ 26 x = − 26 x = − 1 Substitute − 1 for x in equation (2)

7th Edition

ISBN: 9780135902738

Author: Nagle

Publisher: PEARSON

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Differential Equations

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Chapter 12.2, Problem 8E

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The critical point and classification of it for the given system of linear equation.

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Chapter 12.2, Problem 7E

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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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Chapter 12 Solutions

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The critical point and classification of it for the given system of linear equation.