To show that the origin of a given system is an unstable fixed point. To show that all trajectories approach to the ellipse by considering the potential function V= (1- 4x 2 − y 2 ) 2 . Concept Introduction: Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x ˙ is used and x&*#x00A0;is a fixed point Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system. Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points. Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t → ∞ .

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Answer 1

Question

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

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Answer 2

Question

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

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Answer 3

Question

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

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Answer 4

Question

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

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Answer 5

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

Answer 6

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

Answer 7

Chapter 7.3, Problem 4E

Interpretation Introduction

Interpretation:

To show that the origin of a given system is an unstable fixed point.

To show that all trajectories approach to the ellipse by considering the potential function

V= (1- 4x2−y2)2.

Concept Introduction:

Fixed point of a differential equation is a point where, f(x) = 0 ; while substitution f(x) = x˙ is used and x&*#x00A0;is a fixed point

Potential function is a single valued scalar function, the negative gradient of which gives the required function of a system.

Fixed points can be stable or unstable. Phase trajectories go away fromunstable fixed points, while they come towards the stable fixed points.

Limit cycle is the stable trajectory of a system towards which all trajectories eventually come as t→∞.

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.2 - Prob. 1E

Solution Summary: The author explains that the origin of a given system is an unstable fixed point and all trajectories approach to the ellipse by considering the potential function V= (1- 4x

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