Ordinary Differential Equations (MATH-5123)

Introduction

This course  paved the path for differentiation. In many fields of physics,   differential equations are used. We will learn here about preliminaries and classification of differential equations,  Verification of solution, existence of unique solutions,  Introduction to initial value problems, Differential equations as mathematical models, First order ordinary differential equations: Basic concepts, formation and solution of differential equations, Separable equations, Linear equations , integrating factors, Exact Equations,  Solution of some nonlinear first order DEs by substitution, Homogeneous Equations, Bernoulli equation, Ricaati’s equation and Clairaut equation, Modeling with first-order ODEs: Linear models, Nonlinear models,  Higher order differential equations: Initial value and boundary value problems, Homogeneous and Non homogeneous linear DEs and their solutions,Linear dependence and independence, Wronskian, Reduction of order, homogeneous equations with constant coefficient,  Nonhomogeneous equations, undetermined coefficients method, Superposition principle, Annihilator approach, variation of parameters, Cauchy-Euler equation, Solving system of linear DEs by elimination, solution of nonlinear DEs, Series Solutions: Power series, ordinary and singular points, Existence of power series solutions, Solutions about singular points, types of singular points, Frobenius theorem, Existence of Frobenius series solutions, Special functions, The Bessel, Modified Bessel, Legendre and Hermite equations and their solutions. 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

Course Outline:

Introduction, Mathematical Modeling of First and Second Order Differential Equations (ODEs), Solutions and Applications of First Order Differential Equations, Formation and Solutions of Higher Order Linear Differential Equations, Differential Equations with Variable Coefficients, Sturm-Liouville (S-L) System and Boundary-Value Problems, Series Solution and its Limitations, The Frobenius method.

Recommended Books:

1.    Mathematical Methods for Physicists by George Arfken and Hans J. Weber, (6th and onwards editions) Acad Press.  

2.    Differential Equations with boundary-value problems, by  D. G. Zill, M. R. Cullen, PWS Publishing Co. (1997).

3.    Advanced Engineering Mathematics, Erwin Kreyszig, (2007). 

4.    Calculus Early Transcendentals by James Stewart Brooks/Cole (5th and onwards editions)

Assessment Criteria:

Sessional:                    20 marks (Assignment, quiz, etc)

Mid Term exam:           30 marks

Final exam:                  50 marks

Time of class:

BS 3rd (R)      =>       Wednesday (08:00 - 09:00), Thrusday (08:00 - 09:00),Friday(08:00 - 09:00)