Chapter 4: Q16 E (page 212)

Use the energy integral lemma to show that every solution to the Duffing equation (18) is bounded; that is $|y (t)| ⩽ M$ , for some M. [Hint: First argue that for $\frac{y^{2}}{2} + \frac{y^{4}}{4} ⩽ K$ some K.]

Short Answer

Expert verified

Therefore, the given statement is true. The solution to the Duffing equation is bounded and that is, $|y| ⩽ M$ for some M is true.

Step by step solution

General form

The Energy Integral Lemma:

Let y(t) be a solution to the differential equation, $y = f (y)$ where f(y) is a continuous function that does not depend on y’ or the independent variable t. Let F(y) is an indefinite integral of, $f (y)$ that is, $f (y) = \frac{d}{dy} F (y)$ . Then the quantity $E (t) : = \frac{1}{2} y' {(t)}^{2} - F (y (t))$ is constant; i.e. $\frac{d}{dt} E (t) = 0$

Duffing equation: (18)

$y + y + y^{3} = y + (1 + y^{2}) y = 0$ … (1)

Verify the equations

Given that, the Duffing equation is $y + (1 + y^{2}) y = 0$

Then, $y = - (y + y^{3})$ . So, $f (y) = - (y + y^{3})$

Now find the F(y).

$f (y) = \frac{d}{dy} F (y) - (y + y^{3}) = \frac{d}{dy} (- \frac{y^{2}}{2} - \frac{y^{4}}{4}) F (y) = - \frac{y^{2}}{2} - \frac{y^{4}}{4}$

Then, find the quantity.

$E = \frac{1}{2} {(y' (t))}^{2} - F (y (t)) = \frac{1}{2} {(y' (t))}^{2} + \frac{y^{2}}{2} + \frac{y^{4}}{4}$

So, $\frac{1}{2} {(y' (t))}^{2} + \frac{y^{2}}{2} + \frac{y^{4}}{4} = constant$

Verify the equation

Since the above equations are square terms. Then,

$\frac{1}{2} {(y' (t))}^{2} + \frac{y^{2}}{2} + \frac{y^{4}}{4} ⩾ 0 \frac{1}{2} {(y' (t))}^{2} + \frac{y^{4}}{4} ⩾ 0$

And,

$\frac{y^{2}}{2} ⩽ K y^{2} ⩽ 2 K |y| ⩽ \sqrt{2 K} = M$

So, $|y| ⩽ M$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the energy integral lemma to show that every solution to the Duffing equation (18) is bounded; that is $|y (t)| ⩽ M$ , for some M. [Hint: First argue that for $\frac{y^{2}}{2} + \frac{y^{4}}{4} ⩽ K$ some K.]

Short Answer

Step by step solution

General form

Verify the equations

Verify the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Decision Maths

Statistics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the energy integral lemma to show that every solution to the Duffing equation (18) is bounded; that is yt⩽M, for some M. [Hint: First argue that fory22+y44⩽K some K.]

Short Answer

Step by step solution

General form

Verify the equations

Verify the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Decision Maths

Statistics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use the energy integral lemma to show that every solution to the Duffing equation (18) is bounded; that is $|y (t)| ⩽ M$ , for some M. [Hint: First argue that for $\frac{y^{2}}{2} + \frac{y^{4}}{4} ⩽ K$ some K.]