Lecture 1: General Introduction
Lecture Details :
Ordinary Differential Equations and Applications by A. K. Nandakumaran,P. S. Datti & Raju K. George,Department of Mathematics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in.
Course Description :
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 - Preliminaries:Basic concepts from linear algebra - Some important preliminaries from analysis like uniform convergence, Arzela-Ascoli theorem, fixed point theorems etc - 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.
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) - 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 - 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: