Ordinary Differential Equations

AMS-501

Course Syllabus, MS Word format

Course Syllabus, pdf format

Tentative Class Schedule:

Jan. 27: Introduction to AMS501. ODE's; definitions and introductory examples; IVP and BVP.
Jan. 29: Theory of homogeneous linear equations; Wronskian, well-posed and ill-posed problems. Constant coefficient equations.
Feb. 3: Linear homogeneous and inhomogeneous equations. Methods of variation of parameters and undetermined coefficients.
Feb. 5: The method of Green's function.
Feb. 10: Eigenvalue problems; Sturm-Liouville problem.
Feb. 12: First order nonlinear ODE's. Introduction into higher order nonlinear ODE's.
Feb. 17: Systems of linear differential equations. Systems with constant coefficients and non-defective matrices.
Feb. 19: Systems of linear differential equations with non-diagonalizable matrices. Nonhomogeneous systems of linear differential equations.
Feb. 24: Examples of mathematical modeling. Transformation to optimal dependent and independent variables.
Feb. 26: Equations of Lagrangian and Hamiltonian mechanics.
Mar. 3: Review of methods for exactly solvable equations.
Mar. 5: Midterm Exam (in class)
Mar. 10: Review of the midterm exam. Classification of singular points of homogeneous equations.
Mar. 12: Local behavior near ordinary points. Local series expansions about regular singular points.
Mar. 17: Frobenius series for equations with regular singular points.
Mar. 19: Local behavior at irregular singular points of homogeneous linear equations. The method of dominant balance.
Mar. 24: Irregular singular point at infinity.
Mar. 26: Local analysis of inhomogeneous linear equations (illustrative examples).
Mar. 31: Asymptotic analysis of nonlinear equations. Spontaneous singularities.
Apr. 2: Nonlinear autonomous systems. Classification of critical points.
Apr. 14: Critical point analysis of two-dimensional nonlinear systems.
Apr. 16: Regular and singular perturbation theory.
Apr. 23: Asymptotic matching.
Apr. 28: Boundary layer-type perturbation theory.
Apr. 30: Mathematical structure of boundary layers.
May 5: WKB theory.
May 8: Summation of divergent series.
Final Exam (take home)

Homework Assignments and Solutions:

Homework1.                Solutions of Homework1.

Homework 2.               Solutions of Homework 2.

Homework 3.               Solutions of Homework 3.

Homework 4.    

Final exam  Due May 19