Explanation Given: The systems of differential equations: d x d t = − 2 x + 2 x y , d y d t = x − y + x 2 The point ( 0 , 0 ) is the critical point. Approach: 1) Critical points are the solutions of x ′ = 0 and y ′ = 0 . 2) For the autonomous system, x ′ ( t ) = a x + b y + F ( x , y ) y ′ ( t ) = c x + d y + G ( x , y ) where a , b , c , d are constants such that a d − b c ≠ 0 and F , G is continuous in a neighbourhood of the origin. Then origin is the only critical point of the system and the system is almost system is almost linear near origin if F ( x , y ) x 2 + y 2 → 0 and G ( x , y ) x 2 + y 2 → 0 . 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: The systems of differential equations is: d x d t = − 2 x + 2 x y , d y d t = x − y + x 2 The functions F ( x , y ) = 2 x y , G ( x , y ) = x 2 are everywhere continuous. lim ( x , y ) → 0 F ( x , y ) x 2 + y 2 = lim ( x , y ) → 0 2 x y x 2 + y 2 ... ( 1 ) Take x = cos θ , y = sin θ in ( 1 ) .

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 12.3, Problem 2E

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The given system is almost linear near the origin and type and stability of critical point at the origin.

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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 given system is almost linear near the origin and type and stability of critical point at the origin.

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