Explanation Given: y 0 ( t ) has been given as y 0 ( t ) = lim ω → λ ( y ( t ) ) Where, y ( t ) = ( 1 λ 2 − ω 2 ( λ sin ( ω t ) − ω sin ( λ t ) ) ) Formula used: The L’Hopital’s rule was used. When x goes to a particular value, f ( x ) = 0 and g ( x ) = 0 lim x → 0 = f ( x ) g ( x ) = lim x → 0 f ' ( x ) g ' ( x ) Calculation: Using the given data, y ( 0 ) ( t ) = lim ω → λ y ( t ) y ( 0 ) ( t ) = lim ω → λ ( 1 λ 2 − ω 2 ( λ sin ( ω t ) − ω sin ( λ t ) ) ) ( substituting y ( t ) ) = lim ω → λ ( d ( ( λ sin ( ω t ) − ω sin ( λ t ) ) ) To determine (b) When t goes to infinity y 0 ( t ) ceases to be periodic and the value of amplitude | y 0 ( t ) | . To determine (c) To graph: The oscillation of the spring, y ( t ) = ( 1 λ 2 − ω 2 ( λ sin ( ω t ) − ω sin ( λ t ) ) )

4th Edition

ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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Differential Equations

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Question

Chapter 4.5, Problem 80E

To determine

(a)

The value of y0(t) .

To determine

(b)

When t goes to infinity y0(t) ceases to be periodic and the value of amplitude |y0(t)|.

To determine

(c)

To graph:

The oscillation of the spring, y(t)=(1λ2−ω2(λsin(ωt)−ωsin(λt)))

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Chapter 4.5, Problem 79E

Chapter 4.5, Problem 81E

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Chapter 4 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

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(a) The value of y 0 ( t ) .

Solution Summary: The author calculates the value of y_0(t) using the L’Hopital’s rule.

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(a)

The value of y0(t) .

(b)

When t goes to infinity y0(t) ceases to be periodic and the value of amplitude |y0(t)|.

(c)

To graph:

The oscillation of the spring, y(t)=(1λ2−ω2(λsin(ωt)−ωsin(λt)))

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Chapter 4 Solutions

(a) The value of y 0 ( t ) .

Solution Summary: The author calculates the value of y_0(t) using the L’Hopital’s rule.

Concept explainers

Videos

(a) The value of y0(t) .

(b) When t goes to infinity y0(t) ceases to be periodic and the value of amplitude |y0(t)|.

(c) To graph: The oscillation of the spring, y(t)=(1λ2−ω2(λsin(ωt)−ωsin(λt)))

Want to see the full answer?

Chapter 4 Solutions

(a)

The value of y0(t) .

(b)

When t goes to infinity y0(t) ceases to be periodic and the value of amplitude |y0(t)|.

(c)

To graph:

The oscillation of the spring, y(t)=(1λ2−ω2(λsin(ωt)−ωsin(λt)))