(Undamped harmonic oscillator) The first equation that you should study is the harmonic oscillator with no damping; that is, b = 0 and with k = 0. Examine solutions using both their graphs and the phase plane. Are the solutions periodic? If so, what does the period seem to be? Describe the behavior of three different solutions that have especially different initial conditions and be specific about the physical interpretation of the different initial conditions. (Analytic methods to answer these questions are discussed in Chapter 3. For now, work numerically.)
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APPENDIX A
CHANGING VARIABLES
APPENDIX B
POWER SERIES: THE ULTIMATE GUESS
LAB 1.1
Rate of Memorization Model
LAB 1.2
Growth of a Population of Mold
LAB 1.3
Logistic Population Models with Harvesting
LAB 1.4
Exponential and Logistic Population Models
LAB 1.5
Modeling Oil Production
LAB 2.1
Two Magnets and a Spring
LAB 2.2
Cooperative and Competitive Species
LAB 2.3
The Harmonic Oscillator with Modified Damping
LAB 2.4
A Mass-Spring System with a Rubber Band
LAB 2.5
Active Shock Absorbers
LAB 3.1
Bifurcations in Linear Systems
LAB 3.2
RLC Circuits
LAB 3.3
Measuring Mass in Space
LAB 3.4
Exploring a Parameter Space
LAB 3.5
Find Your Own Harmonic Oscillator
LAB 3.6
A Baby Bottle Harmonic Oscillator
LAB 4.1
Two Magnets and a Spring Revisited
LAB 4.2
A Periodically Forced RLC Circuit
LAB 4.3
The Tacoma Narrows Bridge
LAB 5.1
Hard and Soft Springs
LAB 5.2
Higher Order Approximations of the Pendulum
LAB 5.3
A Family of Predator-Prey Equations
LAB 5.4
The Glider
LAB 6.1
Poles
LAB 6.2
Convolutions
LAB 7.1
Errors of Numerical Approximations
LAB 7.2
Lost in Space
LAB 8.1
Newtons Method as a Difference Equation
LAB 8.2
The Delayed Logistic and Two-Dimensional Iteration
LAB 8.3
The Bifurcation Diagram
1
First-Order Differential Equations
1.1
MODELING VIA DIFFERENTIAL EQUATIONS
1.2
ANALYTIC TECHNIQUE: SEPARATION OF VARIABLES
1.3
QUALITATIVE TECHNIQUE: SLOPE FIELDS
1.4
NUMERICAL TECHNIQUE: EULERS METHOD
1.5
EXISTENCE AND UNIQUENESS OF SOLUTIONS
1.6
EQUILIBRIA AND THE PHASE LINE
1.7
BIFURCATIONS
1.8
LINEAR EQUATIONS
1.9
INTEGRATING FACTORS FOR LINEAR EQUATIONS
2
First-Order Systems
2.1
MODELING VIA SYSTEMS
2.2
THE GEOMETRY OF SYSTEMS
2.3
THE DAMPED HARMONIC OSCILLATOR
2.4
ADDITIONAL ANALYTIC METHODS FOR SPECIAL SYSTEMS
2.5
EULERS METHOD FOR SYSTEMS
2.6
Existence and Uniqueness for Systems
2.7
THE SIR MODEL OF AN EPIDEMIC
2.8
THE LORENZ EQUATIONS
3
Linear Systems
3.1
PROPERTIES OF LINEAR SYSTEMS AND THE LINEARITY PRINCIPLE
3.2
STRAIGHT-LINE SOLUTIONS
3.3
PHASE PORTRAITS FOR LINEAR SYSTEMS WITH REAL EIGENVALUES
3.4
COMPLEX EIGENVALUES
3.5
SPECIAL CASES: REPEATED AND ZERO EIGENVALUES
3.6
SECOND-ORDER LINEAR EQUATIONS
3.7
THE TRACE-DETERMINANT PLANE
3.8
LINEAR SYSTEMS IN THREE DIMENSIONS
4
Forcing and Resonance
4.1
FORCED HARMONIC OSCILLATORS
4.2
SINUSOIDAL FORCING
4.3
UNDAMPED FORCING AND RESONANCE
4.4
AMPLITUDE AND PHASE OF THE STEADY STATE
4.5
THE TACOMA NARROWS BRIDGE
5
Nonlinear Systems
5.1
EQUILIBRIUM POINT ANALYSIS
5.2
QUALITATIVE ANALYSIS
5.3
HAMILTONIAN SYSTEMS
5.4
DISSIPATIVE SYSTEMS
5.5
NONLINEAR SYSTEMS IN THREE DIMENSIONS
5.6
PERIODIC FORCING OF NONLINEAR SYSTEMS AND CHAOS
6
Laplace Transforms
6.1
LAPLACE TRANSFORMS
6.2
DISCONTINUOUS FUNCTIONS
6.3
SECOND-ORDER EQUATIONS
6.4
DELTA FUNCTIONS AND IMPULSE FORCING
6.5
CONVOLUTIONS
6.6
THE QUALITATIVE THEORY OF LAPLACE TRANSFORMS
7
Numerical Methods
7.1
NUMERICAL ERROR IN EULERS METHOD
7.2
IMPROVING EULERS METHOD
7.3
THE RUNGE-KUTTA METHOD
7.4
THE EFFECTS OF FINITE ARITHMETIC
8
Discrete Dynamical Systems
8.1
THE DISCRETE LOGISTIC EQUATION
8.2
FIXED POINTS AND PERIODIC POINTS
8.3
BIFURCATIONS
8.4
CHAOS
Textbook Solutions for Differential Equations 00
Chapter LAB 2.3 Problem 4
Question
(Nonlinear second-order equation) Finally, consider a somewhat related second-order equation where the damping coefficient b is replaced by the factor (y2 ); that is, m d2 y dt2 + (y2 )dy dt + ky = 0. Is it reasonable to interpret this factor as some type of damping? Provide a complete description of the long-term behavior of the solutions. Are the solutions periodic? If so, what does the period seem to be? Explain why this equation is not a good model for something like a mass-spring system. Give an example of some other type of physical or biological phenomenon that could be modeled by this equation.
Solution
The first step in solving LAB 2.3 problem number 4 trying to solve the problem we have to refer to the textbook question: (Nonlinear second-order equation) Finally, consider a somewhat related second-order equation where the damping coefficient b is replaced by the factor (y2 ); that is, m d2 y dt2 + (y2 )dy dt + ky = 0. Is it reasonable to interpret this factor as some type of damping? Provide a complete description of the long-term behavior of the solutions. Are the solutions periodic? If so, what does the period seem to be? Explain why this equation is not a good model for something like a mass-spring system. Give an example of some other type of physical or biological phenomenon that could be modeled by this equation.
From the textbook chapter The Harmonic Oscillator with Modified Damping you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Differential Equations 00 4
Author
Paul (Paul Blanchard) Blanchard, Robert L. Devaney, Glen R. Hall
ISBN
9780495561989
(Nonlinear second-order equation) Finally, consider a
Chapter LAB 2.3 textbook questions
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Chapter 0: Problem 1 Differential Equations 00 4
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Chapter 0: Problem 2 Differential Equations 00 4
(Harmonic oscillator with damping) Repeat Part 1 using the equation m d2 y dt2 + b dy dt + ky = 0.
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Chapter 0: Problem 3 Differential Equations 00 4
(Harmonic oscillator with nonlinear damping) Repeat Part 1 using the equation m d2 y dt2 + b dy dt dy dt + ky = 0 in place of the usual harmonic oscillator equation. (Note that even with the same value of the parameter b, the drag forces in this equation and the equation in Part 2 have the same magnitude only for velocity 1. Also, note that the sign of the term dy dt dy dt is the same as the sign of dy/dt, hence this damping force is always directed opposite the direction of motion. The difference between this equation and that in Part 2 is the size of the damping for small and large velocities. One of the many examples of situations for which this is a better model than linear damping is the drag on airplane tires from wet snow or slush. Drag from only four inches of slush was enough to cause the 1958 crash during take-off of the plane carrying the Manchester United soccer team. Currently, large airplanes are allowed to take off and land in no more than one-half inch of wet snow or slush.
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Chapter 0: Problem 4 Differential Equations 00 4
(Nonlinear second-order equation) Finally, consider a somewhat related second-order equation where the damping coefficient b is replaced by the factor (y2 ); that is, m d2 y dt2 + (y2 )dy dt + ky = 0. Is it reasonable to interpret this factor as some type of damping? Provide a complete description of the long-term behavior of the solutions. Are the solutions periodic? If so, what does the period seem to be? Explain why this equation is not a good model for something like a mass-spring system. Give an example of some other type of physical or biological phenomenon that could be modeled by this equation.
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