**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

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