Ordinary Differential Equations

Specific Objectives of course:

To introduce students to the formulation, classification of differential equations and existence and uniqueness of solutions. To provide skill in solving initial value and boundary value problems. To develop understanding and skill in solving first and second order linear homogeneous and non- homogeneous differential equations and solving differential equations using power series methods.

Course Outline:

Preliminaries:

Introduction and formulation, classification of differential equations, existence and uniqueness of solutions, introduction of initial value and boundary value problems

First order ordinary differential equations:

Basic concepts, formation and solution of differential equations. Separable variables, Exact Equations, Homogeneous Equations, Linear equations, integrating factors. Some nonlinear first order equations with known solution, differential equations of Bernoulli and Ricaati type, Clairaut equation, modeling with first-order ODEs, Basic theory of systems of first order linear equations, Homogeneous linear system with constant coefficients, Non homogeneous linear system

Second and higher order linear differential equations:

Initial value and boundary value problems, Homogeneous and non-homogeneous equations, Superposition principle,   homogeneous equations with constant coefficients, Linear independence and Wronskian, Non- homogeneous equations, undetermined coefficients method, variation of parameters, Cauchy-Euler equation, Modeling.

Sturm-Liouville  problems:

Introduction  to  eigen  value  problem, adjoint and self adjoint operators, self adjoint differential equations, eigen  values  and  eigen  functions,  Sturm-Liouville  (S-L)  boundary value problems, regular and singular S-L problems, properties of regular S-L problems

Series  Solutions:

Power  series,  ordinary  and   singular  points, Existence of power series solutions, power series solutions, types of  singular points, Frobenius theorem, Existence of Frobenius series solutions, solutions about singular points, The Bessel, modified Bessel Legendre and Hermite equations and their solutions.