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Notes on Diffy Qs

Differential Equations for Engineers

by Jiří Lebl

(version 5.0)

Typeset in LATEX.

Copyright ©2008–2016 Jiří Lebl

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You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses.

During the writing of these notes, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.

The date is the main identifier of version. The major version / edition number is raised only if there have been substantial changes. Edition number started at 5, that is, version 5.0, as it was not kept track of before.

See http://www.jirka.org/diffyqs/ for more information (including contact information).

Contents
Introduction
 0.1 Notes about these notes
 0.2 Introduction to differential equations
 0.3 Classification of differential equations
1 First order ODEs
 1.1 Integrals as solutions
 1.2 Slope fields
 1.3 Separable equations
 1.4 Linear equations and the integrating factor
 1.5 Substitution
 1.6 Autonomous equations
 1.7 Numerical methods: Euler’s method
 1.8 Exact equations
2 Higher order linear ODEs
 2.1 Second order linear ODEs
 2.2 Constant coefficient second order linear ODEs
 2.3 Higher order linear ODEs
 2.4 Mechanical vibrations
 2.5 Nonhomogeneous equations
 2.6 Forced oscillations and resonance
3 Systems of ODEs
 3.1 Introduction to systems of ODEs
 3.2 Matrices and linear systems
 3.3 Linear systems of ODEs
 3.4 Eigenvalue method
 3.5 Two dimensional systems and their vector fields
 3.6 Second order systems and applications
 3.7 Multiple eigenvalues
 3.8 Matrix exponentials
 3.9 Nonhomogeneous systems
4 Fourier series and PDEs
 4.1 Boundary value problems
 4.2 The trigonometric series
 4.3 More on the Fourier series
 4.4 Sine and cosine series
 4.5 Applications of Fourier series
 4.6 PDEs, separation of variables, and the heat equation
 4.7 One dimensional wave equation
 4.8 D’Alembert solution of the wave equation
 4.9 Steady state temperature and the Laplacian
 4.10 Dirichlet problem in the circle and the Poisson kernel
5 Eigenvalue problems
 5.1 Sturm-Liouville problems
 5.2 Application of eigenfunction series
 5.3 Steady periodic solutions
6 The Laplace transform
 6.1 The Laplace transform
 6.2 Transforms of derivatives and ODEs
 6.3 Convolution
 6.4 Dirac delta and impulse response
7 Power series methods
 7.1 Power series
 7.2 Series solutions of linear second order ODEs
 7.3 Singular points and the method of Frobenius
8 Nonlinear systems
 8.1 Linearization, critical points, and equilibria
 8.2 Stability and classification of isolated critical points
 8.3 Applications of nonlinear systems
 8.4 Limit cycles
 8.5 Chaos
Further Reading
Solutions to Selected Exercises
Index

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