Ordinary Differential Equations and Multivariable Calculus (English)

   Saturday, July 22, 2023   

Starting a new Lecture Notes Series on Ordinary Differential Equations and Multivariable Calculus

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Ordinary Differential Equations and Multivariable Calculus By Lecture Notes together!

Lecture Description of this above Topic:

Lecture 1: Session 1 : What are Differential equations, order, degree and solutions(using GeoGebra) of an ODE.

Lecture 2: Example of differential equation whose order is finite but degree is infinite

Lecture 3: Session 2: Variable separable method using GeoGebra.

Lecture 4: Session 3: Reduction to Variable Separable method. (Two cases)

Lecture 5: Session 4: How to find Half life period, solution of a radioactive substance & radiocarbon dating.

Lecture 6: Session 5: Single Tank Problem: Modelling and Solution using Differential equation.

Lecture 7: Session 6: Murdered Professor ( Application of Differential equations)

Lecture 8: Session 7: Exact Differential Equations along with examples.(See pinned comment)

Lecture 9: Session 8:First order Differential Equation is an Exact Differential equation if and only if My = Nx

Lecture 10: Session 9: Integrating Factor to convert non exact differential equation into Exact differential eq.

Lecture 11: Session 10: Linear Differential equations of first order.

Lecture 12: Session 11:How to reduce Bernoulli DE to linear differential equation?

Lecture 13: Session 12: How to find Orthogonal trajectories using differential equations

Lecture 14: Session 13: Second & Higher order D.E + proof & counter example for superposition principle(Part I)

Lecture 15: Session 14: Initial value problem, basis and General solution of DE via Linear Algebra ( Part II).

Lecture 16: Session 15: Reduction to first order of linear Homogeneous 2nd order differential equation(Part-I).

Lecture 17: Session 16: Reduction to first order for D.E of the from F(y,y',y'')=0 and F(x,y',y'') (Part-II).

Lecture 18: Session 17:Connection between linearly independent/dependent and Wronskian in differential equations

Lecture 19: Session 18: Abel's Formula to find Wronskian without actually knowing the solutions of DE.

Lecture 20: Session 19: Second order Homogeneous Linear differential equation with constant coefficients!!

Lecture 21: Session 20:Examples on higher-order homogeneous linear differential equatn with constant coefficient

Lecture 22: Session 21: Solutions to Second and Higher order Euler Cauchy Differential equations.

Lecture 23: Session 22: Given some functions, how to find differential equation.

Lecture 24: Session 23: Modeling of Undamped Mass Spring system with some examples (Part I).

Lecture 25: Session 24: Modeling and idea on Damped Mass Spring system ( Part II).

Lecture 26: Session 25: Introduction to Non-Homogeneous Ordinary Differential Equations.

Lecture 27: Session 26: Plenty of examples on Method of Undetermined coefficients.

Lecture 28: Session 27: Concept and Examples on Variation of parameters method for second and higher order D.E.

Lecture 29: Session 28: First order linear system of differential equations with constant coefficients.

Lecture 30: Session 29: Examples on all possible cases of eigenvalues & evectors for a 2x2 system of D.E(Part-I)

Lecture 31: Session 30: Examples on all possible cases of eigenvalues & vectors for a 3x3 system of D.E(Part-II)

Lecture 32: Session 31: How to find Real solutions to system of first order linear differential equations.

Lecture 33: Session 32: Examples on Method of undetermined coefficients for System of Differential equations.

Lecture 34: Session 33: Example on Method of Variation of Parameters for System of Differential Equations.

Lecture 35: Session 34: Basics of Electrical circuits and how to solve using differential equations.

Lecture 36: Session 1: What are Laplace Transforms? Linearity property of Laplace transforms.

Lecture 37: Session 2: Plenty of Examples on Laplace Transforms

Lecture 38: Session 3: Laplace inverse is linear whereas Laplace inverse doesn't behave nicely with product.

Lecture 39: Session 4: Concept and examples on functions of exponential growth / order.

Lecture 40: Session 5:When does Laplace of a function exist? A counterexample to existence theorem of Laplace.

Lecture 41: Session 6: Laplace of 1/square root(t) exist but it is not piecewise continuous. Counterexample!!!

Lecture 42: Session 7: Uniqueness of Laplace Transforms and Laplace inverse

Lecture 43: Session 8: First Shifting Theorem in Laplace Transforms--Statement and examples.

Lecture 44: Session 9: What is Laplace of f(at) if Laplace of f(t) is known!!

Lecture 45: Session 10: Laplace Transforms to solve differential equations (Initial Value Problem)

Lecture 46: Session 11: Laplace transform to solve differential equations if INITIAL condition is NOT given at 0

Lecture 47: Session 12: Laplace Transform of INTEGRATION of a function. Examples of Laplace and inverse Laplace.

Lecture 48: Session 13: Unit Step function examples and how to express a function in terms of unit step function

Lecture 49: Session 14:Laplace of Unit step function, Second shifting theorem with examples

Lecture 50: Session 15:Dirac Delta(Impulse) function. Proof of properties like derivative, integration & Laplace

Lecture 51: Session 16: Convolution Theorem. Laplace and Laplace inverse examples using convolution

Lecture 52: Session 17: Solution to Voltera Integral equation using Laplace transform and convolution theorem.

Lecture 53: Session 18: What are periodic functions with examples & Laplace of periodic functions with examples.

Lecture 54: Session 19:Laplace of t^nf(t) is derivative of Laplace of given function f(t) (duality principle).

Lecture 55: Session 20: Laplace Transform of f(t)/t. Integration of Laplace Transforms (Duality principle)

Lecture 56: Session 1: Find and sketch domain of a function for several variables & how to check using GeoGebra!

Lecture 57: Session 2: How to find Level Curves, Imagine Graphs using Level Curves & Finally how to use GeoGebra

Lecture 58: Session 3: Function of several variables is bounded/Unbounded/Open/Closed/ both.

Lecture 59: Session 4: Examples on how to show limit do NOT exist using different paths (several variables)

Lecture 60: Session 5: Geometrical meaning for partial derivatives of a function. Also using GeoGebra.

Lecture 61: Session 6: Chain Rule for function of several variable & how branch diagram help to remember formula

Lecture 62: Session 7: Formula for Derivatives of implicit functions using Chain Rule

Lecture 63: Session 8: Notion of Directional derivatives and its properties followed by examples.

Lecture 64: Session 9: Hessian matrix to find Local maxima, Local minima, Saddle point of a function.

Lecture 65: Session 10: Absolute maxima and minima of a function over closed and bounded domains.

Lecture 66: Session 11: Lagrange's Multiplies method to solve constrained optimization problems.

Lecture 67: Question on Direction Derivative in GATE 2021 exam

Lecture 68: Question on Directional Derivative in GATE 2022 exam

Lecture 69: Question on Directional Derivative in GATE 2019 Exam