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11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275378

Chapter 16, Problem 56RE

Textbook Problem

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Investigation The differential equation

models the oscillating motion of an object on the end of aspring, where *y* is the displacement from equilibrium (positivedirection is downward), measured in feet, *t* is time in seconds, *b* is the magnitude of the resistance to the motion, *k* is the spring constant from Hooke’s Law, and *F*(*t*) is the accelerationimposed on the system.

(a) Solve the differential equation and use a graphing utility tograph the solution for each of the assigned quantities for *b*, *k*, and *F*(*t*).

(b) Describe the effect of increasing the resistance to motion *b*.

(c) Explain how the motion of the object changes when astiffer spring (greater value of *k*) is used.

**a)**

To determine

**To calculate:** The solution of the differential equation, *b*, *k*, *F*(*t*) using graphing utility.

**Given information:**

The given differential equation is

**Concept Used:**

If

**Calculation:**

(i)

Consider the given differential equation is

The coefficients of the differential terms are,

Now putting these in the differential equation

The characteristics equation is

First, we have to find the solution of the characteristics equation

Therefore,

Let

Substituting in the original equation yields

Equating the coefficient of like terms yields

Therefore,

Substituting the given initial conditions,

At,

Now differentiating the general solution,

Substituting the given initial conditions,

At,

Therefore,

Therefore, the general solution of the given differential equation is

**Graph:**

The graph of the differential equation

**ii)** Consider the given differential equation is

The coefficients of the differential terms are,

Now putting these in the differential equation

The characteristics equation is

First, we have to find the solution of the characteristics equation

Therefore,

Let

Substituting in the original equation yields

Equating the coefficient of like terms yields

Therefore,

Substituting the given initial conditions,

At,

Now differentiating the general solution,

Substituting the given initial conditions,

At,

Therefore,

Therefore, the general solution of the given differential equation is

**Graph:**

The graph of the differential equation

**b)**

To determine

The effect of increasing resistance to the motion *b*.

**c)**

To determine

How the motion of the object changes when a stiffer spring (greater value of *k*) is used.

Multivariable Calculus

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