Differential Equations with Mathematica

Preface
1 Introduction to differential equations
1.1 Definitions and concepts
1.2 Solutions of differential equations
1.3 Initial- and boundary-value problems
1.4 Direction fields
1.4.1 Creating interactive applications
2 First-order ordinary differential equations
2.1 Theory of first-order equations: a brief discussion
2.2 Separation of variables
2.3 Homogeneous equations
2.4 Exact equations
2.5 Linear equations
2.5.1 Integrating factor approach
2.5.2 Variation of parameters and the method of undetermined coefficients
2.6 Numerical approximations of solutions to first-order equations
2.6.1 Built-in methods
2.6.2 Other numerical methods
Euler's method
Improved Euler method
The Runge–Kutta method
Application: modeling the spread of a disease
3 Applications of first-order equations
3.1 Orthogonal trajectories
3.2 Population growth and decay
3.2.1 The Malthus model
3.2.2 The logistic equation
3.3 Newton's law of cooling
3.4 Free-falling bodies
4 Higher-order linear differential equations
4.1 Preliminary definitions and notation
4.1.1 Introduction
4.1.2 The nth-order ordinary linear differential equation
4.1.3 Fundamental set of solutions
4.1.4 Existence of a fundamental set of solutions
4.1.5 Reduction of order
4.2 Solving homogeneous equations with constant coefficients
4.2.1 Second-order equations
4.2.2 Higher-order equations
4.3 Introduction to solving nonhomogeneous equations
4.4 Nonhomogeneous equations with constant coefficients: the method of undetermined coefficients
4.4.1 Second-order equations
4.4.2 Higher-order equations
4.5 Nonhomogeneous equations with constant coefficients: variation of parameters
4.5.1 Second-order equations
Summary of variation of parameters for second-order equations
4.5.2 Higher-order nonhomogeneous equations
4.6 Cauchy–Euler equations
4.6.1 Second-order Cauchy–Euler equations
4.6.2 Higher-order Cauchy–Euler equations
4.6.3 Variation of parameters
4.7 Series solutions
4.7.1 Power series solutions about ordinary points
Power series solution method about an ordinary point
4.7.2 Series solutions about regular singular points
4.7.3 Method of Frobenius
Application: zeros of the Bessel functions of the first kind
Application: the wave equation on a circular plate
4.8 Nonlinear equations
5 Applications of higher-order differential equations
5.1 Harmonic motion
5.1.1 Simple harmonic motion
5.1.2 Damped motion
5.1.3 Forced motion
5.1.4 Soft springs
5.1.5 Hard springs
5.1.6 Aging springs
Application: hearing beats and resonance
5.2 The pendulum problem
5.3 Other applications
5.3.1 L-R-C circuits
5.3.2 Deflection of a beam
5.3.3 Bodé plots
5.3.4 The catenary
6 Systems of ordinary differential equations
6.1 Review of matrix algebra and calculus
6.1.1 Defining nested lists, matrices, and vectors
6.1.2 Extracting elements of matrices
6.1.3 Basic computations with matrices
6.1.4 Systems of linear equations
6.1.5 Eigenvalues and eigenvectors
6.1.6 Matrix calculus
6.2 Systems of equations: preliminary definitions and theory
6.2.1 Preliminary theory
6.2.2 Linear systems
6.3 Homogeneous linear systems with constant coefficients
6.3.1 Distinct real eigenvalues
6.3.2 Complex conjugate eigenvalues
6.3.3 Solving initial-value problems
6.3.4 Repeated eigenvalues
6.4 Nonhomogeneous first-order systems: undetermined coefficients, variation of parameters, and the matrix exponential
6.4.1 Undetermined coefficients
6.4.2 Variation of parameters
6.4.3 The matrix exponential
6.5 Numerical methods
6.5.1 Built-in methods
6.5.2 Euler's method
6.5.3 Runge–Kutta method
6.6 Nonlinear systems, linearization, and classification of equilibrium points
6.6.1 Real distinct eigenvalues
6.6.2 Repeated eigenvalues
6.6.3 Complex conjugate eigenvalues
6.6.4 Nonlinear systems
Classification of equilibrium points
7 Applications of systems of ordinary differential equations
7.1 Mechanical and electrical problems with first-order linear systems
7.1.1 L-R-C circuits with loops
7.1.2 L-R-C circuit with one loop
7.1.3 L-R-C circuit with two loops
7.1.4 Spring–mass systems
7.2 Diffusion and population problems with first-order linear systems
7.2.1 Diffusion through a membrane
7.2.2 Diffusion through a double-walled membrane
7.2.3 Population problems
7.3 Applications that lead to nonlinear systems
7.3.1 Biological systems: predator–prey interactions, the Lotka–Volterra system, and food chains in the chemostat
The Lotka–Volterra system
Simple food chain in a chemostat
Long food chain in a chemostat
7.3.2 Physical systems: variable damping
7.3.3 Differential geometry: curvature
8 Laplace transform methods
8.1 The Laplace transform
8.1.1 Definition of the Laplace transform
8.1.2 Exponential order, jump discontinuities, and piecewise continuous functions
8.1.3 Properties of the Laplace transform
8.2 The inverse Laplace transform
8.2.1 Definition of the inverse Laplace transform
Linear factors (nonrepeated)
Repeated linear factors
Irreducible quadratic factors
8.2.2 Laplace transform of an integral
8.3 Solving initial-value problems with the Laplace transform
8.4 Laplace transforms of step and periodic functions
8.4.1 Piecewise defined functions: the unit step function
8.4.2 Solving initial-value problems with piecewise continuous forcing functions
8.4.3 Periodic functions
8.4.4 Impulse functions: the delta function
8.5 The convolution theorem
8.5.1 The convolution theorem
8.5.2 Integral and integrodifferential equations
8.6 Applications of Laplace transforms, Part I
8.6.1 Spring–mass systems revisited
8.6.2 L-R-C circuits revisited
8.6.3 Population problems revisited
Application: the tautochrone
8.7 Laplace transform methods for systems
8.8 Applications of Laplace transforms, Part II
8.8.1 Coupled spring–mass systems
8.8.2 The double pendulum
Application: free vibration of a three-story building
9 Eigenvalue problems and Fourier series
9.1 Boundary-value problems, eigenvalue problems, and Sturm–Liouville problems
9.1.1 Boundary-value problems
9.1.2 Eigenvalue problems
9.1.3 Sturm–Liouville problems
9.2 Fourier sine series and cosine series
9.2.1 Fourier sine series
9.2.2 Fourier cosine series
9.3 Fourier series
9.3.1 Fourier series
9.3.2 Even, odd, and periodic extensions
9.3.3 Differentiation and integration of Fourier series
9.3.4 Parseval's equality
9.4 Generalized Fourier series
10 Partial differential equations
10.1 Introduction to partial differential equations and separation of variables
10.1.1 Introduction
10.1.2 Separation of variables
10.2 The one-dimensional heat equation
10.2.1 The heat equation with homogeneous boundary conditions
10.2.2 Nonhomogeneous boundary conditions
10.2.3 Insulated boundary
10.3 The one-dimensional wave equation
10.3.1 The wave equation
10.3.2 D'Alembert's solution
10.4 Problems in two dimensions: Laplace's equation
10.4.1 Laplace's equation
10.5 Two-dimensional problems in a circular region
10.5.1 Laplace's equation in a circular region
10.5.2 The wave equation in a circular region
Bibliography
Index