Explanation Given: The given plane autonomous system is x ′ = x + x y − 3 x 2 and y ′ = 4 y − 2 x y − y 2 . Calculation: The first equation of given plane autonomous system is as follows: x ′ = x + x y − 3 x 2 ( 1 ) The second equation of given plane autonomous system is as follows: y ′ = 4 y − 2 x y − y 2 ( 2 ) A critical point or stationary point is the point as ( x , y ) which is the solution for the equations x ′ = 0 and y ′ = 0 . Consider x ′ = 0 and y ′ = 0 to obtain the critical points. Substitute 0 for x ′ in the equation ( 1 ) . x ( 1 + y − 3 x ) = 0 ( 3 ) Substitute y ′ = 0 in the equation ( 2 ) . y ( 4 − 2 x − y ) = 0 ( 4 ) Use the zero product property to solve the equation ( 3 ) and equation ( 4 ) . Zero product property states that if product of two or more quantities is zero then one of the quantities must be zero. From equation ( 3 ) the equation obtained by zero product property is as follows: x = 0 ( 5 ) 3 x − y = 1 ( 6 ) From equation ( 4 ) the equation obtained by zero product property is as follows: y = 0 ( 7 ) y + 2 x = 4 ( 8 ) From equation ( 5 ) and equation ( 7 ) it is concluded that the point ( 0 , 0 ) is a critical point because x = 0 and y = 0 . Solve equation ( 5 ) and equation ( 8 ) . Substitute 0 for x in equation ( 8 ) . y + 0 = 4 y = 4 Therefore, the point ( 0 , 4 ) is a critical point. Solve equation ( 7 ) and equation ( 6 ) . Substitute 0 for y in equation ( 6 ) . 3 x − 0 = 1 3 x = 1 x = 1 3 ( Divide by 3 ) Therefore, the point ( 1 3 , 0 ) is a critical point. Solve equation ( 8 ) and equation ( 6 ) . Add equation ( 8 ) and equation ( 6 ) . 2 x + 3 x = 5 5 x = 5 x = 1 ( Divide by 5 ) Substitute 1 for x in equation ( 6 ) . 3 − y = 1 − 3 + y = − 1 ( Multiply by − 1 ) y = 2 ( Add 3 ) Therefore, the point ( 1 , 2 ) is a critical point

Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

9th Edition

ISBN: 9781337604918

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 10, Problem 14RE

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The critical points of plane autonomous system x′=x+xy−3x2 and y′=4y−2xy−y2 and classify these critical points.

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Chapter 10, Problem 13RE

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Bundle: Differential Equations with Boundary-Value Problems, 9th + WebAssign Printed Access Card for Zill's Differential Equations with Boundary-Value Problems, 9th Edition, Single-Term

Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 2E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 4E Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - In Problems 16 write the given nonlinear...Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E

Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - Prob. 15E Ch. 10.1 - In Problems 716 find all critical points of the...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - In Problems 2326 solve the given nonlinear plane...Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - In Problems 916 classify the critical point (0, 0)...Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Determine a condition on the real constant so...Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - In Problems 23-26 a nonhomogeneous linear system...Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - In Problems 310, without solving explicitly,...Ch. 10.3 - Prob. 11E Ch. 10.3 - In Problems 1120 classify (if possible) each...Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Show that the dynamical system x = x + xy y = 1 y...Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Prob. 34E Ch. 10.3 - Prob. 35E Ch. 10.3 - Prob. 36E Ch. 10.3 - When a nonlinear capacitor is present in an...Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Competition Models A competitive interaction is...Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Additional Mathematical Models Damped Pendulum If...Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Prob. 8RE Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - Prob. 11RE Ch. 10 - Discuss the geometric nature of the solutions to...Ch. 10 - Classify the critical point (0, 0) of the given...Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE

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The critical points of plane autonomous system x ′ = x + x y − 3 x 2 and y ′ = 4 y − 2 x y − y 2 and classify these critical points.

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The critical points of plane autonomous system x′=x+xy−3x2 and y′=4y−2xy−y2 and classify these critical points.

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