18.03 Differential Equations | Video Lectures

Lecture 1: The Geometrical View of y'= f(x,y)

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: The Geometrical View of y'= f(x,y): Direction Fields, Integral Curves.

Instructor/speaker: Prof. Arthur Mattuck

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(CC BY-NC-SA)

Lecture 2: Euler's Numerical Method for y'=f(x,y)

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Euler's Numerical Method for y'=f(x,y) and its Generalizations.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 3: Solving First-order Linear ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Solving First-order Linear ODE's; Steady-state and Transient Solutions.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 4: First-order Substitution Methods

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: First-order Substitution Methods: Bernouilli and Homogeneous ODE's.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 5: First-order Autonomous ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: First-order Autonomous ODE's: Qualitative Methods, Applications.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 6: Complex Numbers and Complex Exponentials

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Complex Numbers and Complex Exponentials.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 7: First-order Linear with Constant Coefficients

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 8: Continuation

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 10: Continuation: Complex Characteristic Roots

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 11: Theory of General Second-order Linear Homogeneous ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 12: Continuation: General Theory for Inhomogeneous ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODEs

Instructor/speaker: Prof. Arthur Mattuck

Lecture 13: Finding Particular Solutions to Inhomogeneous ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Finding Particular Sto Inhomogeneous ODEs: Operator and Solution Formulas Involving Exponentials.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 14: Interpretation of the Exceptional Case: Resonance

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Interpretation of the Exceptional Case: Resonance

Instructor/speaker: Prof. Arthur Mattuck

Lecture 15: Introduction to Fourier Series

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Introduction to Fourier Series; Basic Formulas for Period 2(pi)

Instructor/speaker: Prof. Arthur Mattuck

Lecture 16: Continuation: More General Periods

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Continuation: More General Periods; Even and Odd Functions; Periodic Extension.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 17: Finding Particular Solutions via Fourier Series

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 19: Introduction to the Laplace Transform

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Introduction to the Laplace Transform; Basic Formulas

Instructor/speaker: Prof. Arthur Mattuck

Lecture 20: Derivative Formulas

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 21: Convolution Formula

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems

Instructor/speaker: Prof. Arthur Mattuck

Lecture 22: Using Laplace Transform to Solve ODEs with Discontinuous Inputs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Using Laplace Transform to Solve ODEs with Discontinuous Inputs.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 23: Use with Impulse Inputs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions

Instructor/speaker: Prof. Arthur Mattuck

Lecture 24: Introduction to First-order Systems of ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System.

Instructor/speaker: Prof. Arthur Mattuck

Lecture 25: Homogeneous Linear Systems with Constant Coefficients

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case)

Instructor/speaker: Prof. Arthur Mattuck

Lecture 26: Continuation: Repeated Real Eigenvalues

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Continuation: Repeated Real Eigenvalues, Complex Eigenvalues

Instructor/speaker: Prof. Arthur Mattuck

Lecture 27: Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients

Instructor/speaker: Prof. Arthur Mattuck

Lecture 28: Matrix Methods for Inhomogeneous Systems

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters

Instructor/speaker: Prof. Arthur Mattuck

Lecture 29: Matrix Exponentials

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Matrix Exponentials; Application to Solving Systems

Instructor/speaker: Prof. Arthur Mattuck

Lecture 30: Decoupling Linear Systems with Constant Coefficients

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Decoupling Linear Systems with Constant Coefficients

Instructor/speaker: Prof. Arthur Mattuck

Lecture 31: Non-linear Autonomous Systems

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum

Instructor/speaker: Prof. Arthur Mattuck

Lecture 32: Limit Cycles

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Limit Cycles: Existence and Non-existence Criteria

Instructor/speaker: Prof. Arthur Mattuck

Lecture 33: Relation Between Non-linear Systems and First-order ODEs

Miller, Haynes — 2011-03-16T14:26:50+05:00

Topics covered: Relation Between Non-linear Systems and First-order ODEs; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra's Equation and Principle

Instructor/speaker: Prof. Arthur Mattuck