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Differential Equations I

Differential equations are equations that involve an unknown function and its derivatives. They are used to model a wide variety of phenomena in the natural and social sciences. We will introduce the basic concepts of differential equations and explore some simple examples.

Differential Equations I roughly corresponds to a first-year university course on the subject. The main topics covered include first-order differential equations, second-order linear differential equations, and systems of differential equations. We also touch on some partial differential equations.

For more information, this is a rough outline of the topics covered:

  • Classification of differential equations: ODEs, PDEs, order, degree, linearity, homogeneity, autonomy.
  • First-order differential equations: separable equations, linear equations, exact equations, homogeneous equations, Bernoulli equations.
  • Second-order linear differential equations: homogeneous equations with constant coefficients, method of undetermined coefficients, variation of parameters.
  • Power series solutions: Leibniz-Maclaurin method, Frobenius method.
  • Systems of differential equations: matrix methods, eigenvalues and eigenvectors, phase plane analysis. Poincaré diagrams, and approximating nonlinear systems with the Jacobian.
  • Laplace transforms: definition, properties, solving ODEs using Laplace transforms, inverse Laplace transforms (some complex analysis needed). Convolution theorem.
  • Introduction to partial differential equations: classification of second-order linear PDEs (hyperbolic, parabolic, elliptic). Cauchy problems, Hadamard criteria for well-posedness. D-operator notation (endomorphisms in function space).
  • Quasilinear first-order PDEs: method of characteristics
  • Fully nonlinear first-order PDEs: Charpit's method, Hamilton-Jacobi equations.
  • Laplace's equation: separation of variables, Fourier series solutions, boundary value problems. Spherical coordinates (Legendre polynomials, spherical harmonics).
  • Fourier series and transforms: Fourier series representation of periodic functions, Fourier transform definition and properties, solving PDEs using Fourier transforms. Plancherel's theorem, etc.
  • Wave equation: derivation, d'Alembert's solution, separation of variables, Fourier series solutions, initial and boundary value problems.
  • Heat equation: derivation, separation of variables, Fourier series solutions, initial and boundary value problems. Fourier transform methods.
  • Green's functions: definition, properties, constructing Green's functions for ODEs and PDEs, solving inhomogeneous differential equations using Green's functions.
  • Techniques for nonlinear differential equations: perturbation methods, stability analysis, phase plane analysis, bifurcation theory.