Motivation and real life examples: Preliminaries; Basics from linear algebra and real analysis like concepts of dependence, independence, basis, Rank-Nullity theorem, determinants and eigenvalues, remarks on Jordan decomposition theorem - convergence, uniform convergence, fixed point theorems, Lipschitz continuity etc.: First and second order linear equations; Examples, A systematic procedure to solve first order and development of the concept integrating factor, Second order homogeneous and non-homogeneous equations, Wronskian, methods of solving: General Existence and Uniqueness theory; Picard's iteration, Peano's exisentce theory, Existence via Arzela Ascoli theorem, non-uniqueness, continuous dependence: Linear systems; Understanding linear system via linear algebra, stability of Linear systems, Explicit phase portrait in 2D linear with constant coefficients : Periodic Solutions; Stability, Floquet theory, particular case o second order equations-Hill's equation: Sturm-Liouville theory; Oscillation theorems: Qualitative Analysis; Examples of nonlinear systems, Stability analysis, Liapunov stability, phase portrait of 2D systems, Poincare Bendixon theory, Leinard's theorem: Introduction to two-point Boundary value problems; Linear equations, Green's function, nonlinear equations, existence and uniqueness:

Module No.

Topic/s

Lectures

1

Motivation and real life examples:

An introduction about differential equations and why this course.

Present various examples like population growth, spring-mass-dashpot system and other nonlinear system. These examples will be recalled as and when necessary.

4

2

Preliminaries

Basic concepts from linear algebra

Some important preliminaries from analysis like uniform convergence, Arzela-Ascoli theorem, fixed point theorems etc.

5

3

First and second order linear equations

First order linear differential equations, Exact differential equations and integrating factors.

Second order linear differential equations (homogeneous and non-homogeneous. Equation with constant coefficients, analysis of spring-mass-dashpot system.

5

4

General Existence and Uniqueness theory

Examples of non-uniqueness, non-existence, importance of existence uniqueness theory, Picard's iteration, Peano's existence theory, Existence via Arzela Ascoli theorem, continuous dependence:

Methods of solving (series solution).

9

5

Linear systems

Understanding linear system via linear algebra, stability of Linear systems, Explicit phase portrait of 2D linear systems with constant coefficients, General case, Non-homogeneous Systems :

5

6

Qualitative Analysis

Examples of nonlinear systems, Stability analysis, Liapunov stability, phase portrait of 2D systems, Poincare Bendixon theory, Leinard's theorem:

9

7

Introduction to two-point Boundary value problems

Linear equations, Green's function, nonlinear equations, existence and uniqueness:

3

First course on linear algebra and real analysis
(knowledge of multi variable calculus including implicit and inverse function theorems would be preferable)

E. A. Coddington and N. Levinson, Theory of ordinary Differential Equations, Tata-McGraw Hill, 1972.

M. W. Hrisch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems & An Introduction to Chaos, Academic Press, 2004.

E. L. Ince, Ordinary Differential Equations, Dover, 1956.

S. Lefschetz, Differential Equations: Geometric Theory, Dover, 1977.

L. Perko, Differential Equations and Dynamical Systems, Springer International Edition, 2001.

G. F. Simmons, Differential Equations with Applications and Historical Notes, Tata-McGraw Hill, 1991.

G. F. Simmons and S. G. Krantz, Differential Equations; Theory, Techniques and Practice, Tata-McGraw Hill, 2007.

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, Vol. 2, Springer, 1990

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