Ordinary Differential Equations Done Right
Ordinary Differential Equations Done RightThis book should serve as a second course in ODEs for a traditional undergraduate path, or as a first course for advanced undergraduates and early graduate students. Most notably, a full sequence of calculus and linear algebra are necessary prerequisites.
Topics covered include:
Complex numbers and functions
Linear algebra (including Jordan form, matrix functions, and the spectral mapping theorem)
Infinite-dimensional vector spaces, function spaces, and inner product spaces
The Fourier series, Fourier and Laplace transforms
The matrix exponential solution to systems of linear ODEs
Duhamel's principle for nonhomogeneous linear ODEs
Variation of parameters for nonhomogeneous linear ODEs
Nonlinear maps, and linearization
Closed-form solutions to nonlinear ODEs (exact equations and integrating factors)
Euler, Runge-Kutta, and Galerkin methods for approximating solutions
Chaos theory, and the Hartman-Grobman theorem for stability of equilibrium points
The Lie series, and an introduction to PDEs
Both proofs-based, and purely computational exercises are included, with the intent that the course could be administed with-or-without a formal proofs course as a prerequisite.
Furthermore, video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).