111 Use power series to solve the differential equation.y y 0 y
Read moreTable of Contents
1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Graphing Calculators and Computers
1.5
Exponential Functions
1.6
Inverse Functions and Logarithms
2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
The Precise Definition of a Limit
2.5
Continuity
2.6
Limits at Infinity; Horizontal Asymptotes
2.7
Derivatives and Rates of Change
2.8
The Derivative as a Function
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Linear Approximations and Differentials
3.11
Hyperbolic Functions
3.2
The Product and Quotient Rules
3.3
Derivatives of Trigonometric Functions
3.4
The Chain Rule
3.5
Implicit Differentiation
3.6
Derivatives of Logarithmic Functions
3.7
Rates of Change in the Natural and Social Sciences
3.8
Exponential Growth and Decay
3.9
RELATED RATES
4.1
Maximum and Minimum Values
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and lHospitals Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems
4.8
Newtons Method
4.9
Antiderivatives
5.1
Areas and Distances
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
Indefinite Integrals and the Net Change Theorem
6.1
Areas Between Curves
6.2
Volumes
6.3
Volumes by Cylindrical Shells
6.4
Work
6.5
Average Value of a Function
7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables and Computer Algebra Systems
7.7
Approximate Integration
7.8
Improper Integrals
8.1
Arc Length
8.2
Area of a Surface of Revolution
8.3
Applications to Physics and Engineering
8.4
Applications to Economics and Biology
8.5
Probability
9.1
Modeling with Differential Equations
9.2
Direction Fields and Eulers Method
9.3
Separable Equations
9.4
Models for Population Growth
9.5
Linear Equations
9.6
Predator-Prey Systems
10.1
Curves Defined by Parametric Equations
10.2
Calculus with Parametric Curves
10.3
Polar Coordinates
10.4
Areas and Lengths in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates
11.1
Sequences
11.10
Taylor and Maclaurin Series
11.11
Applications of Taylor Polynomials
11.2
Series
11.3
The Integral Test and Estimates of Sums
11.4
The Comparison Tests
11.5
Alternating Series
11.6
Absolute Convergence and the Ratio and Root Tests
11.7
Strategy for Testing Series
11.8
Power Series
11.9
Representations of Functions as Power Series
12.1
Three-Dimensional Coordinate Systems
12.2
Vectors
12.3
The Dot Product
12.4
The Cross Product
12.5
Equations of Lines and Planes
12.6
Cylinders and Quadric Surfaces
13.1
Vector Functions
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration
14.1
Functions of Several Variables
14.2
Limits and Continuity
14.3
Partial Derivatives
14.4
Tangent Planes and Linear Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
14.7
Maximum and Minimum Values
14.8
Lagrange Multipliers
15.1
Double Integrals over Rectangles
15.10
Change of Variables in Multiple Integrals
15.2
Iterated Integrals
15.3
Double Integrals over General Regions
15.4
Double Integrals in Polar Coordinates
15.5
Applications of Double Integrals
15.6
Surface Area
15.7
Triple Integrals
15.8
Triple Integrals in Cylindrical Coordinates
15.9
Triple Integrals in Spherical Coordinates
16.1
Vector Fields
16.10
Summary
16.2
Line Integrals
16.3
The Fundamental Theorem for Line Integrals
16.4
Greens Theorem
16.5
Curl and Divergence
16.6
Parametric Surfaces and Their Areas
16.7
Surface Integrals
16.8
Stokes Theorem
16.9
The Divergence Theorem
17.1
Second-Order Linear Equations
17.2
Nonhomogeneous Linear Equations
17.3
Applications of Second-Order Differential Equations
17.4
Series Solutions
Textbook Solutions for Calculus: Early Transcendentals
Chapter 17.4 Problem 21
Question
Assume that the earth is a solid sphere of uniform density with mass and radius mi. For a particle of mass within the earth at a distance from the earths center, the gravi tational force attracting the particle to the center is where is the gravitational constant and is the mass of the earth within the sphere of radius .(a) Show that .(b) Suppose a hole is drilled through the earth along a diameter.Show that if a particle of mass is dropped from restat the surface, into the hole, then the distance ofthe particle from the center of the earth at time is given bywhere .(c) Conclude from part (b) that the particle undergoes simpleharmonic motion. Find the period T.(d) With what speed does the particle pass through the centerof the earth?
Solution
The first step in solving 17.4 problem number 42 trying to solve the problem we have to refer to the textbook question: Assume that the earth is a solid sphere of uniform density with mass and radius mi. For a particle of mass within the earth at a distance from the earths center, the gravi tational force attracting the particle to the center is where is the gravitational constant and is the mass of the earth within the sphere of radius .(a) Show that .(b) Suppose a hole is drilled through the earth along a diameter.Show that if a particle of mass is dropped from restat the surface, into the hole, then the distance ofthe particle from the center of the earth at time is given bywhere .(c) Conclude from part (b) that the particle undergoes simpleharmonic motion. Find the period T.(d) With what speed does the particle pass through the centerof the earth?
From the textbook chapter Series Solutions you will find a few key concepts needed to solve this.
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Title
Calculus: Early Transcendentals 7
Author
ames Stewart
ISBN
9780538497909
Answer: Assume that the earth is a solid sphere of uniform density with mass and radius
Chapter 17.4 textbook questions
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Chapter 17: Problem 1 Calculus: Early Transcendentals 7
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Chapter 17: Problem 2 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y xy
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Chapter 17: Problem 3 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y x x
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Chapter 17: Problem 4 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.x 3y 2y 0 2 y
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Chapter 17: Problem 5 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y xy y 0 y
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Chapter 17: Problem 6 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y y
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Chapter 17: Problem 7 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.x 1y y 0 y
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Chapter 17: Problem 8 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y xy
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Chapter 17: Problem 9 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y xy y 0 y0 1 y 0 0 y x y0
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Chapter 17: Problem 10 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y x y0 1 y 0 0 2 y 0
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Chapter 17: Problem 11 Calculus: Early Transcendentals 7
111 Use power series to solve the differential equation.y x 2 y xy 0 y0 0 y 0 1 x 2 y
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Chapter 17: Problem 12 Calculus: Early Transcendentals 7
The solution of the initial-value problem is called a Bessel function of order 0. (a) Solve the initial-value problem to find a power series expansion for the Bessel function. ; (b) Graph several Taylor polynomials until you reach one that looks like a good approximation to the Bessel function on the interval .
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Chapter 17: Problem 13 Calculus: Early Transcendentals 7
1114 Solve the initial-value problem.y 5y 4y 0 y0 0 y 0 1 9y y
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Chapter 17: Problem 14 Calculus: Early Transcendentals 7
1114 Solve the initial-value problem.9y y 3x e y0 1 y 0 2 x y 4y
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Chapter 17: Problem 15 Calculus: Early Transcendentals 7
1516 Solve the boundary-value problem, if possible.y 4y 29y 0 y0 1 y 1 y 4y
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Chapter 17: Problem 16 Calculus: Early Transcendentals 7
1516 Solve the boundary-value problem, if possible.y 4y 29y 0 y0 1 y e2 Exercises
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Chapter 17: Problem 17 Calculus: Early Transcendentals 7
Use power series to solve the initial-value problem y xy y 0 y0 0 y 0 1 y 6y
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Chapter 17: Problem 18 Calculus: Early Transcendentals 7
Use power series to solve the equationy xy 2y 0 R
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Chapter 17: Problem 19 Calculus: Early Transcendentals 7
A series circuit contains a resistor with , an inductor with H, a capacitor with F, and a 12-V battery. The initial charge is C and the initial current is 0. Find the charge at time t.
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Chapter 17: Problem 20 Calculus: Early Transcendentals 7
A spring with a mass of 2 kg has damping constant 16, and a force of N keeps the spring stretched m beyond its natural length. Find the position of the mass at time if it starts at the equilibrium position with a velocity of m s.
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Chapter 17: Problem 21 Calculus: Early Transcendentals 7
Assume that the earth is a solid sphere of uniform density with mass and radius mi. For a particle of mass within the earth at a distance from the earths center, the gravi tational force attracting the particle to the center is where is the gravitational constant and is the mass of the earth within the sphere of radius .(a) Show that . (b) Suppose a hole is drilled through the earth along a diameter. Show that if a particle of mass is dropped from rest at the surface, into the hole, then the distance of the particle from the center of the earth at time is given by where . (c) Conclude from part (b) that the particle undergoes simple harmonic motion. Find the period T. (d) With what speed does the particle pass through the center of the earth?
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