Notes On Diffy Qs

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

Contents
Introduction
0.1 Notes about these notes
0.2 Introduction to 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
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
Further Reading
Solutions to Selected Exercises
Index