Overview

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This course provides a comprehensive study of ordinary differential equations (ODEs), focusing on both theoretical concepts and solution techniques. It begins with the classification and solution of linear and non-linear first-order ODEs, followed by an in-depth exploration of second-order ODEs, including the concepts of linear dependence and independence of solutions. Various methods for solving second-order equations are introduced, such as the method of undetermined coefficients, variation of parameters, and reduction of order. The course also covers advanced techniques like the Laplace Transform for solving initial value problems and the application of Fourier series for representing periodic solutions. Sturm-Liouville problems are studied as a framework for solving boundary value problems and understanding eigenfunction expansions. Additionally, the course addresses the solution of systems of linear ODEs using matrix methods and eigenvalue analysis. This course equips students with the mathematical tools necessary to model and analyze real-world dynamic systems in science and engineering.

Syllabus

Introduction to Differential Equations

In this module, differential equations will be introduced. The learner will understand the concept of order, degree of ordinary differential equation, examining various types of solutions, and demonstrating how to form a differential equation from a given solution.

First order Differential Equations

This module introduces various methods for solving first-order ordinary differential equations. Topics covered include separable equations, homogeneous equations and their reducible forms, linear differential equations, Bernoulli's equation, and exact and non-exact differential equations, including the use of integrating factors. The module concludes with illustrative examples.

Applications of first order Differential equations

In this module we will discuss various real world applications which result in ordinary differential equations of first order and also how to solve them by using appropriate methods.

Higher order Linear Differential equations with constant coefficients

In this module,we will discuss higher order linear differential equations with constant coefficients. It consists of homogeneous and non homogeneous types. It covers the method to find particular solutions for different cases.

Laplace Transform Methods

In this module,we will discuss laplace transformation and its properties. Laplace transform of different functions and special functions. We will also discuss Laplace transform of derivatives and integrals

Solution of ODE using Laplace transforms
Introduction to Fourier Series

In this module, you will learn about periodic functions, orthogonality of sine and cosine function. Fourier series is introduced and Euler’s formula to obtain Fourier series of periodic functions with period 2П is derived. Then the convergence of Fourier series is discussed along with some examples. Finally,Gibbs phenomenon is introduced.

Extension and Applications of Fourier Series

In this module, you will understand the concepts of even functions, odd functions. You will be able to find Fourier series of functions with arbitrary periods and simplify the Fourier series of even and odd functions. You will learn to find Fourier series with half range expansions and will be able to apply the concepts for some applications. In this module, Parseval’s identity will be introduced and will be able to apply it.

Sturm Liouville Equation and its Applications

In this module, you will understand the concepts of boundary value problem, eigenvalue and eigenfunction, Sturm Liouville problem. You will be able to apply the orthogonality of eigen functions of Sturm Liouville Problem. Generalized Fourier series will be introduced and will learn about Fourier Legendre series and Fourier Bessel series.

Linear System of ODE

In this module, you will understand the concept of linear system of ODE. The theory of eigenvalue, eigen function and diagonalization of a matrix will be revisited. You will be able to solve the uncoupled linear system using method of separation of variable. Depending on matrix A, you will learn to find the general solution, the linear system of ODE using appropriate methods. Finally, you will be able to apply the concepts to solve initial value problems.