Ordinary Differential Equations

AMS-501

Course Syllabus, MS Word format

Course Syllabus, pdf format

Tentative Class Schedule:

Jan. 26: Introduction to AMS501. ODE's; definitions and introductory examples; IVP and BVP.
Jan. 28: Theory of homogeneous linear equations; Wronskian, well-posed and ill-posed problems. Constant coefficient equations.
Feb. 2: Linear homogeneous and inhomogeneous equations. Methods of variation of parameters and undetermined coefficients.
Feb. 4: The method of Green's function.
Feb. 9: Eigenvalue problems; Sturm-Liouville problem.
Feb. 11: First order nonlinear ODE's. Introduction into higher order nonlinear ODE's.
Feb. 16: Systems of linear differential equations. Systems with constant coefficients and non-defective matrices.
Feb. 18: Systems of linear differential equations with non-diagonalizable matrices. Nonhomogeneous systems of linear differential equations.
Feb. 23: Examples of mathematical modeling. Transformation to optimal dependent and independent variables.
Feb. 25: Equations of Lagrangian and Hamiltonian mechanics.
Mar. 2: Review of methods for exactly solvable equations.
Mar. 4: Midterm Exam (in class)
Mar. 9: Review of the midterm exam. Classification of singular points of homogeneous equations.
Mar. 11: Local behavior near ordinary points. Local series expansions about regular singular points.
Mar. 16: Frobenius series for equations with regular singular points.
Mar. 18: Local behavior at irregular singular points of homogeneous linear equations. The method of dominant balance.
Mar. 23: Irregular singular point at infinity.
Mar. 25: Local analysis of inhomogeneous linear equations (illustrative examples).
Apr. 6: Asymptotic analysis of nonlinear equations. Spontaneous singularities.
Apr. 8: Nonlinear autonomous systems. Classification of critical points.
Apr. 13: Critical point analysis of two-dimensional nonlinear systems.
Apr. 15: Regular and singular perturbation theory.
Apr. 20: Asymptotic matching.
Apr. 22: Boundary layer-type perturbation theory.
Apr. 27: Mathematical structure of boundary layers.
Apr. 29: High order boundary layer theory.
May 4: WKB theory.
May 6: Summation of divergent series.
Final Exam (take home)