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:
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