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Contents

 0.1 Notes about these notes
 0.2 Introduction to differential equations
  0.2.1 Differential equations
  0.2.2 Solutions of differential equations
  0.2.3 Differential equations in practice
  0.2.4 Exercises
1 First order ODEs
 1.1 Integrals as solutions
  1.1.1 Exercises
 1.2 Slope fields
  1.2.1 Slope fields
  1.2.2 Existence and uniqueness
  1.2.3 Exercises
 1.3 Separable equations
  1.3.1 Separable equations
  1.3.2 Implicit solutions
  1.3.3 Examples
  1.3.4 Exercises
 1.4 Linear equations and the integrating factor
  1.4.1 Exercises
 1.5 Substitution
  1.5.1 Substitution
  1.5.2 Bernoulli equations
  1.5.3 Homogeneous equations
  1.5.4 Exercises
 1.6 Autonomous equations
  1.6.1 Exercises
 1.7 Numerical methods: Euler’s method
  1.7.1 Exercises
2 Higher order linear ODEs
 2.1 Second order linear ODEs
  2.1.1 Exercises
 2.2 Constant coefficient second order linear ODEs
  2.2.1 Complex numbers and Euler’s formula
  2.2.2 Complex roots
  2.2.3 Exercises
 2.3 Higher order linear ODEs
  2.3.1 Linear independence
  2.3.2 Constant coefficient higher order ODEs
  2.3.3 Exercises
 2.4 Mechanical vibrations
  2.4.1 Some examples
  2.4.2 Free undamped motion
  2.4.3 Free damped motion
  2.4.4 Exercises
 2.5 Nonhomogeneous equations
  2.5.1 Solving nonhomogeneous equations
  2.5.2 Undetermined coefficients
  2.5.3 Variation of parameters
  2.5.4 Exercises
 2.6 Forced oscillations and resonance
  2.6.1 Undamped forced motion and resonance
  2.6.2 Damped forced motion and practical resonance
  2.6.3 Exercises
3 Systems of ODEs
 3.1 Introduction to systems of ODEs
  3.1.1 Exercises
 3.2 Matrices and linear systems
  3.2.1 Matrices and vectors
  3.2.2 Matrix multiplication
  3.2.3 The determinant
  3.2.4 Solving linear systems
  3.2.5 Computing the inverse
  3.2.6 Exercises
 3.3 Linear systems of ODEs
  3.3.1 Exercises
 3.4 Eigenvalue method
  3.4.1 Eigenvalues and eigenvectors of a matrix
  3.4.2 The eigenvalue method with distinct real eigenvalues
  3.4.3 Complex eigenvalues
  3.4.4 Exercises
 3.5 Two dimensional systems and their vector fields
  3.5.1 Exercises
 3.6 Second order systems and applications
  3.6.1 Undamped mass-spring systems
  3.6.2 Examples
  3.6.3 Forced oscillations
  3.6.4 Exercises
 3.7 Multiple eigenvalues
  3.7.1 Geometric multiplicity
  3.7.2 Defective eigenvalues
  3.7.3 Exercises
 3.8 Matrix exponentials
  3.8.1 Definition
  3.8.2 Simple cases
  3.8.3 General matrices
  3.8.4 Fundamental matrix solutions
  3.8.5 Approximations
  3.8.6 Exercises
 3.9 Nonhomogeneous systems
  3.9.1 First order constant coefficient
  3.9.2 First order variable coefficient
  3.9.3 Second order constant coefficients
  3.9.4 Exercises
4 Fourier series and PDEs
 4.1 Boundary value problems
  4.1.1 Boundary value problems
  4.1.2 Eigenvalue problems
  4.1.3 Orthogonality of eigenfunctions
  4.1.4 Fredholm alternative
  4.1.5 Application
  4.1.6 Exercises
 4.2 The trigonometric series
  4.2.1 Periodic functions and motivation
  4.2.2 Inner product and eigenvector decomposition
  4.2.3 The trigonometric series
  4.2.4 Exercises
 4.3 More on the Fourier series
  4.3.1 2L -periodic functions
  4.3.2 Convergence
  4.3.3 Differentiation and integration of Fourier series
  4.3.4 Rates of convergence and smoothness
  4.3.5 Exercises
 4.4 Sine and cosine series
  4.4.1 Odd and even periodic functions
  4.4.2 Sine and cosine series
  4.4.3 Application
  4.4.4 Exercises
 4.5 Applications of Fourier series
  4.5.1 Periodically forced oscillation
  4.5.2 Resonance
  4.5.3 Exercises
 4.6 PDEs, separation of variables, and the heat equation
  4.6.1 Heat on an insulated wire
  4.6.2 Separation of variables
  4.6.3 Insulated ends
  4.6.4 Exercises
 4.7 One dimensional wave equation
  4.7.1 Exercises
 4.8 D’Alembert solution of the wave equation
  4.8.1 Change of variables
  4.8.2 D’Alembert’s formula
  4.8.3 Another way to solve for the side conditions
  4.8.4 Exercises
 4.9 Steady state temperature and the Laplacian
  4.9.1 Exercises
 4.10 Dirichlet problem in the circle and the Poisson kernel
  4.10.1 Laplace in polar coordinates
  4.10.2 Series solution
  4.10.3 Poisson kernel
  4.10.4 Exercises
5 Eigenvalue problems
 5.1 Sturm-Liouville problems
  5.1.1 Boundary value problems
  5.1.2 Orthogonality
  5.1.3 Fredholm alternative
  5.1.4 Eigenfunction series
  5.1.5 Exercises
 5.2 Application of eigenfunction series
  5.2.1 Exercises
 5.3 Steady periodic solutions
  5.3.1 Forced vibrating string.
  5.3.2 Underground temperature oscillations
  5.3.3 Exercises
6 The Laplace transform
 6.1 The Laplace transform
  6.1.1 The transform
  6.1.2 Existence and uniqueness
  6.1.3 The inverse transform
  6.1.4 Exercises
 6.2 Transforms of derivatives and ODEs
  6.2.1 Transforms of derivatives
  6.2.2 Solving ODEs with the Laplace transform
  6.2.3 Using the Heaviside function
  6.2.4 Transfer functions
  6.2.5 Transforms of integrals
  6.2.6 Exercises
 6.3 Convolution
  6.3.1 The convolution
  6.3.2 Solving ODEs
  6.3.3 Volterra integral equation
  6.3.4 Exercises
 6.4 Dirac delta and impulse response
  6.4.1 Rectangular pulse
  6.4.2 The delta function
  6.4.3 Impulse response
  6.4.4 Three-point beam bending
  6.4.5 Exercises
7 Power series methods
 7.1 Power series
  7.1.1 Definition
  7.1.2 Radius of convergence
  7.1.3 Analytic functions
  7.1.4 Manipulating power series
  7.1.5 Power series for rational functions
  7.1.6 Exercises
 7.2 Series solutions of linear second order ODEs
  7.2.1 Exercises
 7.3 Singular points and the method of Frobenius
  7.3.1 Examples
  7.3.2 The method of Frobenius
  7.3.3 Bessel functions
  7.3.4 Exercises
8 Nonlinear systems
 8.1 Linearization, critical points, and equilibria
  8.1.1 Autonomous systems and phase plane analysis
  8.1.2 Linearization
  8.1.3 Exercises
 8.2 Stability and classification of isolated critical points
  8.2.1 Isolated critical points and almost linear systems
  8.2.2 Stability and classification of isolated critical points
  8.2.3 The trouble with centers
  8.2.4 Conservative equations
  8.2.5 Exercises
 8.3 Applications of nonlinear systems
  8.3.1 Pendulum
  8.3.2 Predator-prey or Lotka-Volterra systems
  8.3.3 Exercises
 8.4 Limit cycles
  8.4.1 Exercises
 8.5 Chaos
  8.5.1 Duffing equation and strange attractors
  8.5.2 The Lorenz system
  8.5.3 Exercises