Differential Equations Laboratory Workbook

Resource originally made available by CODEE on September 19, 2010

Originally published in 1992, this 368-page workbook includes a plethora of classroom ready exercises and projects for students in differential equations courses. Using an approach which closely parallels what goes on in science and engineering laboratories, this workbook provides computer experiments that amplify topics found in introductory ordinary differential equations texts. Excellent 2- and 3-D graphics illustrate the range of qualitative behavior of solutions and give compelling visual evidence of theoretical deductions and a greater understanding of the qualitative properties. The experiments are largely self-contained and are independent of any particular hardware/software platform or text.

This book is made available on this web site with permission from John Wiley and Sons, Hoboken, NJ.

Download Link: DifferentialEquationsLaboratoryWorkbook.pdf

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- Introduction: Learning About Your Hardware/Software
- Solution Curves and Numerical Methods
- First Order Equations
- Second Order Equations
- Planar Systems
- Higher Dimensional Systems
- Appendices

- A Team Laboratory Reports (Some hints on writing team reports)
- B Mathematical Modeling (Derivation of the differential equations that appear in the modeling experiments throughout the workbook)
- B.l Population and Rate Models
- B.2 Mechanics
- B.3 Chemical Reactions
- B.4 Circuits
- B.5 Scaling and Dimensionless Variables
- C The Atlas

Plotting Orbits and Solution Curves

1.1 Direction Fields and Solution Curves

1.2 ODEs in Non Normal Form

1.3 Direction Fields and Orbits

1.4 Solvers and IVPs

Generating Atlas Plots

1.5 Generating Atlas Plots

First Order Rate Laws

1.6 Population Growth and Decay

1.7 Radioactive Decay: Carbon-14 Dating

1.8 Potassium-Argon Dating

Falling Bodies

1.9 Falling Bodies Near the Earth's Surface

1.10 Escape Velocities

Aliasing and Other Phenomena

1.11 Aliasing and Other Phenomena

Properties of Orbits and Solution Curves

2.1 Fundamentals

2.2 Sign Analysis

Equilibrium Solutions and Sensitivity

2.3 Pitchfork Bifurcation

Solutions That Escape to Infinity

2.4 Solutions That Escape to Infinity

Picard Process for Solving IVPs

2.5 Picard Process for Solving IVPs

Euler's Method for Solving IVPs

2.6 Euler's Method and Explicit Solutions

2.7 Limitations of Euler's Method

Euler Solutions to the Logistic Equation

2.8 Convergent Euler Sequences

2.9 Period Doubling and Chaos: Graphical Evidence

2.10 Period Doubling and Chaos: Theory

Linear First Order ODEs: Properties of Solutions

3.1 Superposition

3.2 Singularities

Linear First Order ODEs: Data

3.3 Dependence on Data

3.4 Bounded Input/Bounded Output

Separable ODEs: Implicit Solutions

3.5 Separable ODEs: Implicit Solutions

Nonlinear ODEs: Homogeneous Functions

3.6 Nonlinear ODEs: Homogeneous Functions

The ODE: M(x, y)dx + N(x, y)dy = 0

3.7 Planar Systems and M dx + N dy = 0

3.8 Construction of Integral Curves: The Cat

ODEs in Polar Coordinates

3.9 ODEs in Polar Coordinates

Comparison of Solutions of Two ODEs

3.10 Comparison of Solutions of Two ODEs

Harvesting a Species

3.11 Constant Rate Harvesting

3.12 Variable Rate Harvesting

3.13 Intermittent Harvesting

Salt Levels in a Brine

3.14 Linear Brine Models

3.15 Nonlinear Brine Models

Bimolecular Chemical Reactions

3.16 Quadratic Rates as Bimolecular Models

3.17 Modeling a Bimolecular Reaction

Properties of Solutions

4.1 Properties of Solutions

Constant Coefficient Linear ODEs: Undriven

4.2 Constant Coefficient Linear ODEs: Undriven

Constant Coefficient Linear ODEs: Driven

4.3 Beats and Resonance

4.4 General Driving Forces

Frequency Response Modeling

4.5 Parameter Identification

4.6 Gain and Phase Shift

Springs

4.7 Linear Springs

4.8 Hard and Soft Springs

4.9 Aging Springs

Circuits

4.10 Simple RLC Circuit

4.11 Tuning a Circuit

Portraits of Planar Systems

5.1 Portraits of Planar Systems

Autonomous Linear Systems

5.2 Gallery of Pictures

5.3 Stability

Driven Linear Systems

5.4 Driven Linear Systems

Interacting Species

5.5 Predator-Prey Models: Harvesting, Overcrowding

5.6 Competing Species

The Pendulum

5.7 The Undriven Pendulum: Linear Model

5.8 The Undriven Nonlinear Pendulum

5.9 The Driven Upended Pendulum

Duffing's Equation

5.10 Duffing's Equation

Planetary Motion

5.11 Elements of Orbital Mechanics

Stability and Lyapunov Functions

5.12 The Effect of a Perturbation

5.13 Stability and Lyapunov Functions

Cycles and Limit Cycles

5.14 The van der Pol System

5.15 Systems with Cycles and Limit Cycles

The Poincaré-Bendixson Alternatives

5.16 The Poincaré-Bendixson Alternatives

The Hopf Bifurcation

5.17 The Hopf Bifurcation

5.18 Satiable Predation: Bifurcation to a Cycle

Undriven Linear Systems

6.1 Portraits of Undriven Linear Systems

6.2 Asymptotic Behavior and Eigenelements

Driven Linear Systems

6.3 Transients, Steady States, Resonance

6.4 Coupled Oscillators

A Compartment Model: Lead in the Body

6.5 A Compartment Model: Lead in the Body

Lorenz System: Sensitivity

6.6 Inducing Chaos

6.7 Search for Cycles

Rössler System: Period-doubling

6.8 Rössler System: Sensitivity

The Rotational Stability of a Tennis Racket

6.9 The Rotational Stability of a Tennis Racket

Nonlinear Systems and Chemical Reactions

6.10 Approach to Equilibrium: Five Species

6.11 Approach to Equilibrium: Four Species

Oscillating Chemical Reactions

6.12 On/Off Oscillations: Autocatalator

6.13 Persistent Oscillations: The Oregonator

Bifurcations and Chaos in a Nonlinear Circuit

6.14 Bifurcations and Chaos in a Nonlinear Circuit