Linear Differential Equations and Oscillators (Mathematics and Physics for Science and Technology)

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Diagrams, Notes, and Tables
Series Preface
Preface to Volume IV
Acknowledgments
About the Author
Mathematical Symbols
Physical Quantities
1. Three Cases of Characteristic Polynomials
	1.1. Equation of Order N and Initial Conditions
		1.1.1. An Ordinary Differential Equation and its Integrals
		1.1.2. Arbitrary Constants in the General Integral
		1.1.3. Single- or Multi-Point or Mixed Boundary Values
		1.1.4. Example of a General Integral and of Initial Values
		1.1.5. Families of Integral Curves in the Plane
		1.1.6. Regular and Singular Points of a Differential Equation
		1.1.7. Singular Points of Two Kinds
		1.1.8. Envelope as an Example of a Special Integral
		1.1.9. Differential Equations Satisfied by a Function
		1.1.10. Parametric Differential Equations and Bifurcations
		1.1.11. Single Solution around Stable Equilibrium
		1.1.12. Three Solutions around Stable/Unstable Equilibria
		1.1.13. Poincaré (1892) Diagrams and Parametric Evolution
	1.2. General, Particular, and Complete Integrals
		1.2.1. Linear Unforced and Forced Differential Equations
		1.2.2. Superposition of Linearly Independent Solutions
		1.2.3. Wronskian (1812) and Linear Independence of Differentiable Functions
		1.2.4. General Integral of the Linear Unforced Differential Equation (Lagrange, 1765; Fuchs, 1866)
		1.2.5. A Particular Integral of the Linear Forced Equation
		1.2.6. Complete Integral of the Linear Forced Equation (D’Alembert, 1762)
	1.3. Unforced Linear Equation with Constant Coefficients
		1.3.1. Distinct Roots of the Characteristic Polynomial (Euler, 1743; Cauchy, 1827)
		1.3.2. Linear Independence of the Exponentials and Van der Monde Determinant
		1.3.3. Two Real Distinct Roots and Hyperbolic Functions
		1.3.4. Complex Conjugate Roots and Circular Functions
		1.3.5. Circular and Hyperbolic Functions for Fourth-Order Equations
		1.3.6. Root of Multiplicity M and Variation of Constants
		1.3.7. Parametric Differentiation of Particular Integrals (D’Alembert, 1748)
		1.3.8. Multiple Roots of the Characteristic Polynomial
		1.3.9. Linear Independence of Powers with Fixed Base
		1.3.10. Multiple Pairs of Complex Conjugate or Real Roots
		1.3.11. Characteristic Polynomial with Single and Multiple Roots
		1.3.12. Products of Powers and Circular or Hyperbolic Functions as Particular Integrals
		1.3.13. Symmetric Real and Complex Conjugate Pairs of Roots of the Characteristic Polynomial
		1.3.14. Single and Multiple Roots of the Characteristic Polynomial
	1.4. General (Complete) Integral of the Unforced (Forced) Equation
		1.4.1. Parametric Differentiation of a Forced Solution (D’Alembert, 1762)
		1.4.2. Rule Applied to a Forced Solution (L’Hôspital, 1696; Bernoulli, 1691)
		1.4.3. Resonant and Non-Resonant Forcing by an Exponential
		1.4.4. Forcing by a Hyperbolic Cosine or Sine
		1.4.5. Forcing by a Circular Sine or Cosine
		1.4.6. Product of an Exponential and a Hyperbolic Function
		1.4.7. Product of an Exponential and a Circular Function
		1.4.8. Product of an Exponential and a Circular and a Hyperbolic Function
	1.5. Polynomial of Derivatives and Inverse Operator
		1.5.1. Direct and Inverse Binomial Series
		1.5.2. Interpretation of the Inverse Polynomial of Derivatives
		1.5.3. Partial Fractions for the Inverse Characteristic Polynomial
		1.5.4. Single and Multiple Poles of the Inverse Characteristic Polynomial
		1.5.5. Particular Integral for Polynomial Forcing
		1.5.6. Trial Solution and Determination of Coefficients
		1.5.7. Series of Derivatives Applied to a Smooth Function
		1.5.8. Forcing by the Product of a Smooth Function and an Exponential
		1.5.9. Products of Smooth and Elementary Functions
		1.5.10. Forcing by Products of Elementary Functions
		1.5.11. Distinction between Resonant and Non-Resonant Cases
		1.5.12. Zero as a Root of the Characteristic Polynomial
		1.5.13. Forcing by the Product of a Polynomial and Exponential
		1.5.14. Product of a Power by a Hyperbolic Cosine or Sine
		1.5.15. Product of a Power by a Circular Cosine or Sine
		1.5.16. Product of a Power by an Exponential and a Hyperbolic Function
		1.5.17. Product of a Power by an Exponential by a Circular Function
		1.5.18. Product of a Power by an Exponential and Circular and Hyperbolic Functions
		1.5.19. Forced Differential Equation with Constant Coefficients
	1.6. Homogenous Linear Differential Equation with Power Coefficients (Euler, 1769)
		1.6.1. Transformation into a Linear Equation with Constant Coefficients
		1.6.2. Simple Roots of the Characteristic Polynomial
		1.6.3. Multiple Roots of the Characteristic Polynomial
		1.6.4. General Integral of the Homogenous Differential Equation
		1.6.5. Relation between Ordinary and Homogenous Derivatives
		1.6.6. Real Distinct and Complex Conjugate Roots
		1.6.7. Isotropic Multidimensional Laplace Equation
		1.6.8. Linear, Logarithmic, and Power-Law Potentials
		1.6.9. A Second-Order Linear Homogeneous Differential Equation
		1.6.10. Third-Order Linear Homogeneous Differential Equation
	1.7. Homogeneous Derivatives and Characteristic Polynomial
		1.7.1. Homogeneous Differential Equation Forced by a Power
		1.7.2. Forcing by a Hyperbolic Cosine or Sine of a Logarithm
		1.7.3. Forcing by the Circular Cosine or Sine of a Logarithm
		1.7.4. Product of a Power and a Hyperbolic Function of a Logarithm
		1.7.5. Product of a Power and Circular Functions of a Logarithm
		1.7.6. Product of a Power by Hyperbolic and Circular Functions of a Logarithm
	1.8. Inverse Polynomial of Homogeneous Derivatives
		1.8.1. Series Expansion for the Inverse of a Polynomial
		1.8.2. Homogeneous Derivatives of a Polynomial of Logarithms
		1.8.3. Solution by Substitution by the Inverse Characteristic Polynomial
		1.8.4. Homogeneous Equation Forced by a Polynomial of Logarithms
		1.8.5. Linear Differential Equations with Constant/Homogeneous Coefficients
		1.8.6. Polynomial of Logarithms Multiplied by a Smooth Function
		1.8.7. Product of a Power by the Power of a Logarithm
		1.8.8. Product of a Power by a Hyperbolic Function of a Logarithm
		1.8.9. Product of a Power by a Circular Function of a Logarithm
		1.8.10. Powers and Logarithms Multiplied by Hyperbolic Functions
		1.8.11. Powers and Logarithms Multiplied by Circular Functions
		1.8.12. Product of Power by Hyperbolic and Circular Functions of a Logarithm
		1.8.13. Superposition of Unforced/Forced Particular Integrals
	1.9. Solution of Finite Difference Equations
		1.9.1. Classification of Finite Difference Equations
		1.9.2. Comparison of Three Characteristic Polynomials
		1.9.3. General Solutions of the Unforced Equations
		1.9.4. Particular Solutions of the Forced Equations
		1.9.5. Pair of Complex Conjugate Roots
		1.9.6. Second-Order Finite Difference Equation
		1.9.7. Unforced Biquadratic Finite Difference Equation
		1.9.8. Forced Biquadratic Finite Difference Equation
		1.9.9. Four Initial Conditions for a Finite Difference Equation
		1.9.10. Solutions of the Three Equations with Characteristic Polynomials
	Conclusion 1
2. The Damped/Amplified and Forced Oscillator
	2.1. Linear Electromechanical Circuits
		2.1.1. Mass, Motion, and Inertia Force (Newton, 1686)
		2.1.2. Spring, Restoring Force, and Potential Energy
		2.1.3. Damper, Friction Force, and Dissipation Function
		2.1.4. Kinetic Energy, Dissipation, and Work
		2.1.5. Conservation of the Total Energy
		2.1.6. Electrical Circuit: Self, Resistor, and Capacitor
		2.1.7. Electric and Magnetic Energies and Dissipation (Joule)
		2.1.8. Free Motion and Forced Response
		2.1.9. Unsteady Electromagnetic Field (Maxwell, 1863)
		2.1.10. Decoupled Electro-Magnetostatic Fields and Potentials
		2.1.11. Propagation and Diffusion of Electromagnetic Waves
		2.1.12. Retarded Time and Quasi-Stationary Approximation
		2.1.13. Electrical Circuit with Electromotive Force or Battery
	2.2. Stable Attractor and Unstable Repeller
		2.2.1. Harmonic Oscillator: Mass and Spring
		2.2.2. Oscillation about a Center of Stable Attraction
		2.2.3. Adiabatic Invariant for Slow Evolution
		2.2.4. Stable (Unstable) Suspended (Inverted) Pendulum
		2.2.5. Unstable Repeller and Saddle Point
	2.3. Free Motion with Critical or Supercritical Damping
		2.3.1. Trajectories of an Oscillator with Critical Damping
		2.3.2. Critically Damped Oscillator in the Phase Plane
		2.3.3. Trajectories of an Oscillator with Supercritical Damping
		2.3.4. Supercritically Damped Oscillator in the Phase Plane
	2.4. Free Oscillations with Strong, Weak, or No Damping
		2.4.1. Oscillator with Subcritical Damping
		2.4.2. Strong, Weak, or Negligible Damping
		2.4.3. Average of a Function of Time over a Period of Oscillation
		2.4.4. Equipartition, Conservation, and Dissipation of Energy
		2.4.5. Oscillator with Subcritical Damping in the Phase Plane
	2.5. Mechanical and Electrical Amplified Oscillations
		2.5.1. Amplified Mechanical Circuit
		2.5.2. Zero or Reduced Effective Damping
		2.5.3. Monotonic or Oscillatory Amplification and Overstability
		2.5.4. Pendulum Supported on a Cam with Friction (Froude Pendulum)
		2.5.5. Electrical Circuit with a Valve
	2.6. Comparison of Zero with Small Mass
		2.6.1. Uniform, Damped, or Amplified Motion
		2.6.2. Proximity of Initial Conditions and Condition of Small Mass
	2.7. Forced Oscillations, Resonance, and Beats
		2.7.1. Applied Frequency Distinct from Natural Frequency
		2.7.2. Forced Oscillations with Constant Amplitude and Energy
		2.7.3. Applied Frequency Equal to the Natural Frequency
		2.7.4. Undamped Resonance with Growing Amplitude and Energy
		2.7.5. Beats for Close Applied and Natural Frequencies
		2.7.6. Low-Frequency Envelope and Energy Modulation for Beats
	2.8. Damped Resonance for the Frequency and Decay
		2.8.1. Sinusoidal Forcing in the Presence of Damping
		2.8.2. Amplitude and Phase of Forced Oscillations
		2.8.3. Constant Amplitude and Balance of Activity and Damping
		2.8.4. Simultaneous Forcing in Frequency and Damping
		2.8.5. Coincidence of One or Both of Frequency and Decay
		2.8.6. Resonance with Damping and Decaying Forcings
	2.9. Arbitrary, Bounded, and Exponential Forcings
		2.9.1. Response to an Integrable Forcing Function
		2.9.2. Method of Variation of Parameters for Forcing
		2.9.3. Arbitrary Forcing with Critical or Non-Critical Damping
		2.9.4. Cases of Exponential and Constant Forcing
		2.9.5. Upper Bound for the Displacement Due to Finite Forcing
	Conclusion 2
Bibliography
Index