Explanation Given: The systems of differential equations: d x d t = − 2 x + y , d y d t = − 2 y The point ( 0 , 0 ) is the critical point. Approach: 1) Critical points are the solutions of x ′ = 0 and y ′ = 0 . 2) Trajectories are the parametric solutions of the system of differential equations. 3) Eigenvalues are the roots of characteristic equation. 4) Classification of critical points: i) For complex conjugates eigenvalues: When the real part λ is zero, this type of critical point is called a center. It is neutrally stable. When the real part λ is nonzero, this type of critical point is called a spiral point. It is asymptotically stable if real part is less than 0 , it is unstable if real part is greater than 0 . ii) For real eigenvalues: Distinct eigenvalues When both eigenvalues are positive or both are negative, this type of critical point is called a node. It is asymptotically stable if eigenvalues are both negative, unstable if eigenvalues are both positive. When eigenvalues have opposite signs, this type of critical point is called a saddle point. It is always unstable. Repeated eigenvalues When there are two linearly independent eigenvectors, this type of critical point is called a proper node. It is asymptotically stable if eigenvalue is less than 0 and unstable if eigenvalue is greater than 0 . When there is only one linearly independent eigenvector, this type of critical point is called an improper node. It is asymptotically stable if eigenvalue is less than 0 , unstable if eigenvalue is greater than 0 . Calculation: Write the given system in matrix form. d X = [ − 2 1 0 − 2 ] X . Here, X = [ x y ] Find the eigenvalue of the matrix [ − 2 1 0 − 2 ]

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 12.2, Problem 19E

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The type of critical point at the origin and sketch the phase plane diagram for the system.

dxdt=−2x+y,dydt=−2y

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Chapter 12.2, Problem 18E

Chapter 12.2, Problem 20E

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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 type of critical point at the origin and sketch the phase plane diagram for the system. d x d t = − 2 x + y , d y d t = − 2 y

Solution Summary: The author explains the critical point at the origin and the phase plane diagram for the system.

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The type of critical point at the origin and sketch the phase plane diagram for the system.

dxdt=−2x+y,dydt=−2y

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Chapter 12 Solutions

The type of critical point at the origin and sketch the phase plane diagram for the system. d x d t = − 2 x + y , d y d t = − 2 y

Solution Summary: The author explains the critical point at the origin and the phase plane diagram for the system.

Videos

The type of critical point at the origin and sketch the phase plane diagram for the system. dxdt=−2x+y,dydt=−2y

Want to see the full answer?

Chapter 12 Solutions

The type of critical point at the origin and sketch the phase plane diagram for the system.

dxdt=−2x+y,dydt=−2y