**Ordinary Differential
Equations**

**AMS-501**

## 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)