Explanation Given: The differential equation is y ′ ( t ) = y 3 − y 2 − 2 y . Definition used: “If f ( y ) < 0 on the left of a and f ( y ) > 0 on the right of a , then the equilibrium solution y = a is unstable .” “If f ( y ) > 0 on the left of a and f ( y ) < 0 on the right of a , then the equilibrium solution y = a is stable .” Calculation: Obtain the equilibrium solution by setting y ′ ( t ) = 0 . y 3 − y 2 − 2 y = 0 y ( y 2 − y − 2 ) = 0 Simplify the equation, y = 0 And y 2 − y − 2 = 0 y 2 − 2 y + y − 2 = 0 y ( y − 2 ) + 1 ( y − 2 ) = 0 ( y − 2 ) ( y + 1 ) = 0 That is, y = 2 and y = − 1 . Therefore, the equilibrium solution are y = 0 , y = 2 and y = − 1 . The direction field for the equation y ′ ( t ) = y 3 − y 2 − 2 y is shown in below Figure 1. From Figure 1, it is observed that the equilibrium point y = 0 is stable and the points y = 2 and y = − 1 is unstable. First to check the equilibrium solution y = 0 is stable or unstable. Let y ′ ( t ) = f ( y ) . f ( y ) = y 3 − y 2 − 2 y Take any value of y on the left of 0 and obtain f ( y ) . f ( − 1 ) = ( − 1 ) 3 − ( − 1 ) 2 − 2 ( − 1 ) = − 1 − 1 + 2 > 0 That is, f ( y ) > 0 when the value of y on the left of 0. Take any value of y on the right of 0 and obtain f ( y )

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 9, Problem 25RE

To determine

To find: The equilibrium solution of the given equation and check whether it is stable or unstable.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9, Problem 24RE

Chapter 9, Problem 26RE

Students have asked these similar questions

Determine the equilibrium solutions, if any, for the given equation. Identify equilibrium solutions as stable, unstable or semi-stable.

Determine whether the following PDE is equilibrium equation or dynamical equation. du Ət dx² dy² а. cannot be determined b. equilibrium equation С. dynamical equation

Consider the autonomous equation: P = 0.05P(100 – P)-h, where P is an animal population, and h is the number of animals harvested per unit time. Determine the values of h which yield no equilibrium states, one equilibrium state, and two equilibrium states. (Your responses should involve equalities or inequalities involving h and a real number.) No equilibrium state: One semi-stable equilibrium state: Two distinct equilibrium states:

Chapter 9 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 9.1 - What are the orders of the equations in Example 2?...Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - In Example 7, if the height function were given by...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - The solution to the initial value problem y(t) = 2...

Knowledge Booster

$Calculus$

Learn more about

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

The equilibrium solution of the given equation and check whether it is stable or unstable.

Solution Summary: The author explains how to find the equilibrium solution of the given equation and check whether it is stable or unstable.

Concept explainers

Videos

To find: The equilibrium solution of the given equation and check whether it is stable or unstable.

Want to see the full answer?

Chapter 9 Solutions