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

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Videos

Question

Chapter 10, Problem 14RE

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10, Problem 13RE

Chapter 10, Problem 15RE

Students have asked these similar questions

Evaluate the triple integral 3' 23 HIG 2 +3 f(x, y, z)dxdydz where f(x, y, z) = x + 2x-y ม u = v = and w = 2 2 3 Triple Integral Region R -2 x N 2 y 3

Find the volume of the solid bounded below by the circular cone z = 2.5√√√x² + y² and above by the sphere x² + y²+z² = 6.5z.

Electric charge is distributed over the triangular region D shown below so that the charge density at (x, y) is σ(x, y) = 4xy, measured in coulumbs per square meter (C/m²). Find the total charge on D. Round your answer to four decimal places. 1 U 5 4 3 2 1 1 2 5 7 coulumbs

Chapter 10 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

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

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

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.

Solution Summary: The author explains the critical points of plane autonomous system xprime and classifies them.

Concept explainers

Videos

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

Want to see the full answer?

Chapter 10 Solutions