Screencast Lectures on Ordinary Differential Equations and Applications

Course Description

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.

Screencast Lectures on Ordinary Differential Equations and Applications
Screenshot from Lecture 25: 2-by-2 systems and Phase Plane Analysis
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Video Lectures & Study Materials

Visit the official course website for more study materials:

# Lecture Play Lecture
I. Motivation and Real Life Examples
1 General Introduction to ODEs (59:59) Play Video
2 Examples of ODEs (58:28) Play Video
3 Examples of ODEs II (1:00:26) Play Video
4 Examples of ODEs III (59:41) Play Video
II. Preliminaries
5 Review of Linear Algebra (52:32) Play Video
6 Review of Linear Algebra II (57:56) Play Video
7 Review of Linear Algebra III (1:04:12) Play Video
8 Analysis (1:02:18) Play Video
9 Analysis II (55:16) Play Video
III. First and Second Order Linear Equations
10 First Order Linear Equations (1:00:31) Play Video
11 Exact Equations (1:00:43) Play Video
12 Second Order Linear Equations (59:37) Play Video
13 Second Order Linear Equations II (1:00:25) Play Video
14 Second Order Linear Equations III (59:31) Play Video
IV. General Existence and Uniqueness Theory
15 Wellposedness and Examples of IVP (1:01:51) Play Video
16 Gronwall's Lemma (59:34) Play Video
17 Basic Lemma and Uniqueness Theorem (56:55) Play Video
18 Picard's Existence and Uniqueness Theorem (58:39) Play Video
19 Picard's Existence and Uniqueness Theorem II (58:25) Play Video
20 Cauchy Peano Existence Theorem (59:27) Play Video
21 Existence using Fixed Point Theorem (59:51) Play Video
22 Continuation of Solutions (1:00:17) Play Video
23 Series Solution (1:00:13) Play Video
V. Linear Systems
24 General System and Diagonalizability (58:23) Play Video
25 2-by-2 systems and Phase Plane Analysis (1:02:02) Play Video
26 2-by-2 systems and Phase Plane Analysis II (1:00:07) Play Video
27 General Systems (1:00:39) Play Video
28 General Systems II and Non-homogeneous Systems (1:03:43) Play Video
VI. Nonlinear Systems
29 Basic Definitions and Examples (57:15) Play Video
30 Stability Equilibrium Points (59:57) Play Video
31 Stability Equilibrium Points II (54:05) Play Video
32 Stability Equilibrium Points III (1:00:16) Play Video
33 Second Order Linear Equations IV (58:43) Play Video
34 Lyapunov Function (58:39) Play Video
35 Lyapunov Function II (51:35) Play Video
36 Periodic Orbits and Poincare Bendixon Theory (1:00:28) Play Video
37 Periodic Orbits and Poincare Bendixon Theory II (41:38) Play Video
VII. Two-point Boundary Value Problems
38 Linear Second Order Equations (55:02) Play Video
39 General Second Order Equations (51:02) Play Video
40 General Second Order Equations II (54:17) Play Video


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