Ordinary Differential Equations

AMS-501


Course Syllabus, pdf format


Tentative Class Schedule:

Lecture 1: Introduction to AMS501. ODE's; definitions and introductory examples; IVP and BVP.
Lecture 2: Theory of homogeneous linear equations; Wronskian, well-posed and ill-posed problems. Constant coefficient equations.
Lecture 3: Linear homogeneous and inhomogeneous equations. Methods of variation of parameters and undetermined coefficients.
Lecture 4: The method of Green's function.
Lecture 5: Eigenvalue problems; Sturm-Liouville problem.
Lecture 6:  First order nonlinear ODE's. Introduction into higher order nonlinear ODE's.
Lecture 7: Systems of linear differential equations. Systems with constant coefficients and non-defective matrices.
Lecture 8: Systems of linear differential equations with non-diagonalizable matrices. Nonhomogeneous systems of linear differential equations.
Lecture 9: Examples of mathematical modeling. Transformation to optimal dependent and independent variables.
Lecture 10: Equations of Lagrangian and Hamiltonian mechanics.
Lecture 11: Review of methods for exactly solvable equations.
Mar. 4: Midterm Exam (in class)
Lecture 12: Review of the midterm exam. Classification of singular points of homogeneous equations.
Lecture 13: Local behavior near ordinary points. Local series expansions about regular singular points.
Lecture 14: Frobenius series for equations with regular singular points.
Lecture 15: Local behavior at irregular singular points of homogeneous linear equations. The method of dominant balance.
Lecture 16: Irregular singular point at infinity.
Lecture 17: Local analysis of inhomogeneous linear equations (illustrative examples).
Lecture 18: Asymptotic analysis of nonlinear equations. Spontaneous singularities.
Lecture 19: Nonlinear autonomous systems. Classification of critical points.
Lecture 20: Critical point analysis of two-dimensional nonlinear systems.
Lecture 21: Regular and singular perturbation theory.
Lecture 22: Asymptotic matching.
Lecture 23: Boundary layer-type perturbation theory.
Lecture 24: Mathematical structure of boundary layers.
Lecture 25: High order boundary layer theory.
Lecture 26: WKB theory.
Lecture 27: Summation of divergent series.
Final Exam (in class)