To show that the z-axis is an invariant line for the Lorenz equations. Concept Introduction: ➢ The Lorenz equations are given as x ˙ = σ ( y - x ) , y ˙ = rx - xz - y , z ˙ = xy - bz Here, σ, r, b > 0 .

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Answer 1

Question

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

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Solve the initial value problem: y' = x3 + y³ 3 , y(1) = 2 xy² y(x) = Hint: Notice that the equation on the right is homogeneous and see Homework exercise 23 in section 1.2 of our textbook to review techniques for solving homogeneous equations. Note that we've been given an intial value of the form y(a) = b where a > 0, so this only determines a solution corresponding to the right half of the graph of In(x), i.e., the part of the graph corresponding to positive values of x. Therefore, we should write In(x) instead of ln(|x|), since the left half of the graph is not determined by the initial condition given.

Answer 2

Question

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Answer 6

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Answer 7

Chapter 9.2, Problem 4E

Interpretation Introduction

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Answer 8

Expert Solution & Answer

Answer 9

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Answer 10

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Answer 11

Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.2 - Prob. 1E Ch. 9.2 - Prob. 2E Ch. 9.2 - Prob. 3E Ch. 9.2 - Prob. 4E Ch. 9.2 - Prob. 5E

To show that the z-axis is an invariant line for the Lorenz equations. Concept Introduction: ➢ The Lorenz equations are given as x ˙ = σ ( y - x ) , y ˙ = rx - xz - y , z ˙ = xy - bz Here, σ, r, b > 0 .

Videos

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.

Want to see the full answer?

Chapter 9 Solutions

To show that the z-axis is an invariant line for the Lorenz equations. Concept Introduction: ➢ The Lorenz equations are given as x ˙ = σ ( y - x ) , y ˙ = rx - xz - y , z ˙ = xy - bz Here, σ, r, b &gt; 0 .

Videos

Interpretation: To show that the z-axis is an invariant line for the Lorenz equations. Concept Introduction: ➢ The Lorenz equations are given as x˙=σ(y - x), y˙=rx - xz - y, z˙=xy - bz Here, σ, r, b > 0.

Want to see the full answer?

Chapter 9 Solutions

To show that the z-axis is an invariant line for the Lorenz equations. Concept Introduction: ➢ The Lorenz equations are given as x ˙ = σ ( y - x ) , y ˙ = rx - xz - y , z ˙ = xy - bz Here, σ, r, b > 0 .

Interpretation:

To show that the z-axis is an invariant line for the Lorenz equations.

Concept Introduction:

➢ The Lorenz equations are given as

x˙=σ(y - x),

y˙=rx - xz - y,

z˙=xy - bz

Here, σ, r, b > 0.