Introduction to differential equations with dynamical systems

Introduction to differential equations with dynamical systems / Stephen L. Campbell and Richard Haberman.

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Bibliographic Details
Main Author: Campbell, S. L. (Stephen La Vern)
Other Authors: Haberman, Richard, 1945-
Format: Book
Language:English
Published: Princeton, N.J. : Princeton Univ. Press, c2008.
Subjects:
Differentiable dynamical systems.
Differential equations.
Online Access:Table of contents
Contributor biographical information
Publisher description
Table of Contents:
  • First-Order Differential Equations and Their Applications
  • Introduction to ordinary differential equations
  • The definite integral and the initial value problem
  • The initial value problem and the indefinite integral
  • The initial value problem and the definite integral
  • Mechanics I: elementary motion of a particle with gravity only
  • First-order separable differential equations
  • Using definite integrals for separable differential equations
  • Direction fields
  • Existence and uniqueness
  • Euler's numerical method (optional)
  • First-order linear differential equations
  • Form of the general solution
  • Solutions of homogeneous first-order linear differential equations
  • Integrating factors for first-order linear differential equations
  • Linear first-order differential equations with constant coefficients and constant input
  • Homogeneous linear differential equations with constant coefficients
  • Constant coefficient linear differential equations with constant input
  • Constant coefficient differential equations with exponential input
  • Constant coefficient differential equations with discontinuous input
  • Growth and decay rroblems
  • A first model of population growth
  • Radioactive decay
  • Thermal cooling
  • Mixture problems
  • Mixture problems with a fixed volume
  • Mixture problems with variable volumes
  • Electronic circuits
  • Mechanics II: including air Rrsistance
  • Orthogonal trajectories (optional)
  • Linear Second- and Higher-Order Differential Equations
  • General solution of second-order linear differential equations
  • Initial value problem (for homogeneous equations)
  • Reduction of order
  • Homogeneous linear constant coefficient differential equations (second order)
  • Homogeneous linear constant coefficient differential equations (nth-order)
  • Mechanical vibrations I: formulation and free response
  • Formulation of equations
  • Simple harmonic motion (no damping, 8 = 0)
  • Free response with friction (8 > 0)
  • The method of undetermined coefficients
  • Mechanical vibrations II: forced response
  • Friction is absent (8 = 0)
  • Friction is present (8 > 0) (damped forced oscillations)
  • Linear electric circuits
  • Euler equation
  • Variation of parameters (second-order)
  • Variation of parameters (nth-order)
  • The Laplace Transform
  • Definition and basic properties
  • The shifting theorem (multiplying by an exponential)
  • Derivative theorem (multiplying by t)
  • Inverse laplace transforms (roots, quadratics, and partial fractions)
  • Initial value problems for differential equations
  • Discontinuous forcing functions
  • Solution of differential equations
  • Periodic functions
  • Integrals and the convolution theorem
  • Derivation of the convolution theorem (optional)
  • Impulses and distributions
  • An Introduction to Linear Systems of Differential Equations and Their Phase Plane
  • Introduction
  • Introduction to linear systems of differential equations
  • Solving linear systems using eigenvalues and eigenvectors of the matrix
  • Solving linear systems if the eigenvalues are real and unequal
  • Finding general solutions of linear systems in the case of complex eigenvalues
  • Special systems with complex eigenvalues (optional)
  • General solution of a linear system if the two real eigenvalues are equal (repeated) roots
  • Eigenvalues and trace and determinant (optional)
  • The phase plane for linear systems of differential equations
  • Introduction to the phase plane for linear systems of differential equations
  • Phase plane for linear systems of differential equations
  • Real eigenvalues
  • Complex eigenvalues
  • General theorems
  • Mostly Nonlinear First-Order Differential Equations
  • First-order differential equations
  • Equilibria and stability
  • Equilibrium
  • Stability
  • Review of linearization
  • Linear stability analysis
  • One-dimensional phase lines
  • Application to population dynamics: the logistic equation
  • Nonlinear Systems of Differential Equations in the Plane
  • Introduction
  • Equilibria of nonlinear systems, linear stability analysis of equilibrium, and the phase plane
  • Linear stability analysis and the phase plane
  • Nonlinear systems: summary, philosophy, phase plane, direction field, nullclines
  • Population models
  • Two competing species
  • Predator-prey population models
  • Mechanical systems
  • Nonlinear pendulum
  • Linearized pendulum
  • Conservative systems and the energy integral
  • The phase plane and the potential
  • Answers to odd-numbered exercises.

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