Q16 E Use the energy integral lemma to... [FREE SOLUTION]

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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

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 $|y (t)| 猢� M$ , for some M. [Hint: First argue that for $\frac{y^{2}}{2} + \frac{y^{4}}{4} 猢� K$ some K.]

Short Answer

Expert verified

Therefore, the given statement is true. The solution to the Duffing equation is bounded and that is, $|y| 猢� M$ 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鈥� 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)

$y + y + y^{3} = y + (1 + y^{2}) y = 0$ 鈥� (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,

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

And,

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

So, $|y| 猢� M$

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91影视

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

Short Answer

Step by step solution

General form

Verify the equations

Verify the equation

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

Geometry

Pure Maths

Statistics

Decision Maths

Calculus

Applied Mathematics

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 $|y (t)| 猢� M$ , for some M. [Hint: First argue that for $\frac{y^{2}}{2} + \frac{y^{4}}{4} 猢� K$ some K.]