### Ordinary Differential Equations (MATH-5123)

Description

An ordinary differential equation (ODE) is an equation that involves some ordinary derivatives (as opposed to partial derivatives) of a function. Often, our goal is to solve an ODE, i.e., determine what function or functions satisfy the equation. In general, solving an ODE is more complicated than simple integration. Even so, the basic principle is always integration, as we need to go from derivative to function. Usually, the difficult part is determining what integration we need to do.

Contents

1 Introduction to differential equations: preliminaries and classification of differential equations
2 Verification of solution, existence of unique solutions
3 Introduction to initial value problems
4 Differential equations as mathematical models
5 First order ordinary differential equations: Basic concepts, formation and solution of differential equations
6 Separable equations, Linear equations , integrating factors, Exact Equations
7 Solution of some nonlinear first order DEs by substitution, Homogeneous Equations, Bernoulli equation, Ricaati’s equation and Clairaut equation
8 Modeling with first-order ODEs: Linear models, Nonlinear models
9 Higher order differential equations: Initial value and boundary value problems, Homogeneous and Non homogeneous linear DEs and their solutions,Linear dependence and independence, Wronskian
10 Reduction of order, homogeneous equations with constant coefficients
11 Nonhomogeneous equations, undetermined coefficients method, Superposition principle, Annihilator approach, variation of parameters, Cauchy-Euler equation
12 Solving system of linear DEs by elimination, solution of nonlinear DEs
13 Series Solutions: Power series, ordinary and singular points, Existence of power series solutions

14 Solutions about singular points, types of singular points, Frobenius theorem, Existence of Frobenius series solutions,
15 Special functions, The Bessel, Modified Bessel, Legendre and Hermite equations and their solutions.
16 Sturm-Liouville problems: Introduction to eigen value problem, adjoint and self adjoint operators
17 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

Recommended Books

1. Zill, D.G., and Michael, R., Differential equations with boundary-value problems by Cullin, 5th ed. (Brooks/Cole, 1997).
2. Boyce, W.E. and Diprima,R.C. Elementary differential equations and boundary value problems, 7th ed. (John Wiley & Sons, Inc.)

Suggested Books
1. Arnold, V.I., Ordinary Differential Equations, (Springer, 1991)
2. Apostol, T., Multi Variable Calculus and Linear Algebra, 2nd ed. (John Wiley and sons, 1997).

ASSESSMENT CRITERIA

Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

Mid-Term Exam:  30

Final-Term Exam: 50

### Key Dates and Time of Class Meeting

Wednesday                                                           2.00pm-3:00pm

Thursday                                                               3:00pm-4:00pm

Friday                                                                    2:00pm-3:00pm

Commencement of Classes                                                   October 12, 2020

Mid Term Examination                                                            December 14-18, 2020

Final Term Examination                                                          February 08-12, 2021

Declaration of Result                                                              February 19, 2021