The Nature of Differential Equations: Separable Equations
Introduction
General Remarks on Solutions
Families of Curves: Orthogonal Trajectories
Growth, Decay, Chemical Reactions, and Mixing
Falling Bodies and Other Motion Problems
Brachistochrone: Fermat and the Bernoullis
Miscellaneous Problems for Chapter 1
Appendix: Some Ideas from the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation
First-Order Equations
Homogeneous Equations
Exact Equations
Integrating Factors
Linear Equations
Reduction of Order
Hanging Chain: Pursuit Curves
Simple Electric Circuits
Miscellaneous Problems for Chapter 2
Second-Order Linear Equations
Introduction
General Solution of the Homogeneous Equation
Use of a Known Solution to Find Another
Homogeneous Equation with Constant Coefficients
Method of Undetermined Coefficients
Method of Variation of Parameters
Vibrations in Mechanical and Electrical Systems
Newton’s Law of Gravitation and the Motion of the Planets
Higher-Order Linear Equations: Coupled Harmonic Oscillators
Operator Methods for Finding Particular Solutions
Appendix: Euler
Appendix: Newton
Qualitative Properties of Solutions
Oscillations and the Sturm Separation Theorem
Sturm Comparison Theorem
Power Series Solutions and Special Functions
Introduction: A Review of Power Series
Series Solutions of First-Order Equations
Second-Order Linear Equations: Ordinary Points
Regular Singular Points
Regular Singular Points (Continued)
Gauss’s Hypergeometric Equation
Point at Infinity
Appendix: Two Convergence Proofs
Appendix: Hermite Polynomials and Quantum Mechanics
Appendix: Gauss
Appendix: Chebyshev Polynomials and the Minimax Property
Appendix: Riemann’s Equation
Fourier Series and Orthogonal Functions
Fourier Coefficients
Problem of Convergence
Even and Odd Functions: Cosine and Sine Series
Extension to Arbitrary Intervals
Orthogonal Functions
Mean Convergence of Fourier Series
Appendix: A Pointwise Convergence Theorem
Partial Differential Equations and Boundary Value Problems
Introduction: Historical Remarks
Eigenvalues, Eigenfunctions, and the Vibrating String
Heat Equation
Dirichlet Problem for a Circle: Poisson’s Integral
Sturm–Liouville Problems
Appendix: Existence of Eigenvalues and Eigenfunctions
Some Special Functions of Mathematical Physics
Legendre Polynomials
Properties of Legendre Polynomials
Bessel Functions: The Gamma Function
Properties of Bessel Functions
Appendix: Legendre Polynomials and Potential Theory
Appendix: Bessel Functions and the Vibrating Membrane
Appendix: Additional Properties of Bessel Functions
Laplace Transforms
Introduction
Few Remarks on the Theory
Applications to Differential Equations
Derivatives and Integrals of Laplace Transforms
Convolutions and Abel’s Mechanical Problem
More about Convolutions: The Unit Step and Impulse
Functions
Appendix: Laplace
Appendix: Abel
Systems of First-Order Equations
General Remarks on Systems
Linear Systems
Homogeneous Linear Systems with Constant Coefficients
Nonlinear Systems: Volterra’s Prey–Predator Equations
Nonlinear Equations
Autonomous Systems: The Phase Plane and Its Phenomena
Types of Critical Points: Stability
Critical Points and Stability for Linear Systems
Stability by Liapunov’s Direct Method
Simple Critical Points of Nonlinear Systems
Nonlinear Mechanics: Conservative Systems
Periodic Solutions: The Poincaré–Bendixson Theorem
More about the Van Der Pol Equation
Appendix: Poincaré
Appendix: Proof of Liénard’s Theorem
Calculus of Variations
Introduction: Some Typical Problems of the Subject
Euler’s Differential Equation for an Extremal
Isoperimetric Problems
Appendix: Lagrange
Appendix: Hamilton’s Principle and Its Implications
The Existence and Uniqueness of Solutions
Method of Successive Approximations
Picard’s Theorem
Systems: Second-Order Linear Equation
Numerical Methods
(by John S. Robertson)
Introduction
Method of Euler
Errors
An Improvement to Euler
Higher-Order Methods
Systems
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