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AMS 501, Differential Equations and Boundary Value Problems I
Examples of initial and boundary value problems in which differential equations arise. Existence and uniqueness of solutions, systems of linear differential equations, and the fundamental solution matrix. Power series solutions, Sturm-Louiville theory, eigenfunction expansion, Green's functions.
3 credits, ABCF grading
This course is offered in the Fall semester
Text (required) : "Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory" by Carl Bender and Steven Orszag; Springer; October 29, 1999; ISBN #9780387989310
: "Elementary Differential Equations with Boundary Value Problems" by Edwards & Penney,
6th edition; Pearson Publishing; e-Book; ISBN: 9780321998095
1.) First Order Ordinary Differential Equations:
* Existence and uniqueness, slope field;
* Separable equations, exact equation, reducible equations;
* Linear first order equation;
* Substitution method, Bernoulli equation, etc;
* System of first order equations.
2.) Second Order Ordinary Differential Equations:
* Properties of solutions to linear homogeneous equation;
* Linear homogeneous equation with constant coefficients;
* Non-homogeneous equations;
* Linear equation with variable coefficients;
* Laplace transform, engineering applications.
3.) Stability Theory of Ordinary Differential Equations:
* Stability and phase plane;
* Linear and almost linear system;
* Ecological system, limit cycle;
* Nonlinear mechanical system, dynamic system and chaos.
4.) Boundary Value Problems:
* Power series solutions at regular and regular-singular point;
* Sturm-Liouville problem;
* Eigenvalue and eigenfunctions;
* Cylindrical coordinate problem, Bessel equation;
* Spherical coordinate problem, Legendre equation.