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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Course Curriculum

General Introduction Details 59:59
Examples Details 58:28
Examples Continued I Details 1:26
Examples Continued II Details 59:41
Linear Algebra Details 52:32
Linear Algebra Continued I Details 57:56
Linear Algebra Continued II Details 1:4:12
Analysis Details 1:2:18
Analysis Continued I Details 55:16
First Order Linear Equations Details 1:31
Exact Equations Details 1:43
Second Order Linear Equations Details 59:37
Second Order Linear Equations Continued I Details 1:25
Second Order Linear Equations Continued II Details 59:31
Well-posedness and Examples of IVP Details 1:1:51
Gronwall’s Lemma Details 59:34
Basic Lemma and Uniqueness Theorem Details 56:55
Picard’s Existence and Uniqueness Theorem Details 58:39
Picard’s Existence and Uniqueness I Details 58:25
Cauchy Peano Existence Theorem Details 59:27
Existence using Fixed Point Theorem Details 59:51
Continuation of Solutions Details 1:17
Series Solution Details 1:13
General System and Diagonalizability Details 58:23
2 by 2 systems and Phase Plane Analysis Details 1:2:2
2 by 2 systems and Phase Plane Analysis Continued I Details 1:7
General Systems Details 1:39
General Systems Continued and Non-homogeneous Systems I Details 1:3:43
Basic Definitions and Examples Details 57:15
Stability Equilibrium Points Details 59:57
Stability Equilibrium Points Continued I Details 54:5
Stability Equilibrium Points Continued II Details 1:16
Second Order Linear Equations Continued III Details 58:43
Lyapunov Function Details 58:39
Lyapunov Function I Details 51:35
Periodic Orbits and Poincare Bendixon Theory Details 1:28
Periodic Orbits and Poincare Bendixon Theory I Details 41:38
Linear Second Order Equations Details 55:2
General Second Order Equations Details 51:2
General Second Order Equations Continued I Details 54:17

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