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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 35RE
Textbook Problem
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Motion of a SpringIn Exercise 35-36, a 64-pound weight stretched a spring 4 3 foot from its natural length. Use the given information to find a formula for the position of the weight as a function of time.

The weight is pulled 1 2 foot below equilibrium and released.

To determine

The formula for the position of weight as a function of time when a 64-pounds weight stretches a spring 43 foot from its natural length. The weight is pulled 12 foot below the equilibrium and released.

Explanation of Solution

Given information: The weight is pulled 12 foot below the equilibrium and released.

The 64-pounds weight stretches a spring 43 foot from its natural length, so by Hooke’s Law,

64=k(43)64×34=k48=k

As the weight w is given by mg, we have

m=wg=6432=2

So, the resulting differential equation for the undamped motion is

d2ydt2+482y=0d2ydt2+24y=0

The characteristic equation for the differential equation d2ydt2+24y=0 is m2+24=0.

The roots of the characteristic equation m2+24=0 are:

m2+24=0m2=24m=24m=±26i

If m1=α+iβ and m2=αiβ are complex zeros of the characteristics equation, then the general solution is y=C1eαtcosβt+C2eαtsinβt.

Thus, the general solution of the equation d2ydt2+24y=0 is y=C1cos26t+C2sin26t

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Chapter 16 Solutions

Multivariable Calculus
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Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Graphical and Analytic Analysis In Exercises 15...Ch. 16.1 - Graphical and Analytic AnalysisIn Exercises 15 and...Ch. 16.1 - Finding a Particular Solution In Exercises17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Integrating Factor Show that each expression is...Ch. 16.1 - Integrating FactorShow that thedifferential...Ch. 16.1 - Tangent Curves In Exercises 39-42, use agraphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Finding an Equation of a Curve In Exercise 43 and...Ch. 16.1 - Finding an Equation of a Curve In Exercises 43 and...Ch. 16.1 - Cost In a manufacturing process where y=C(x)...Ch. 16.1 - HOW DO YOU SEE? 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