MIT 18.03 Differential Equations - Spring 2006
67 Lectures · Feb 5, 2003
About
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams.
Course Homepage 18.03 Differential Equations Spring 2006
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Lecture 1: The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves
Jan 19, 2009
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Lecture 2: Euler's Numerical Method for y'=f(x,y) and its Generalizations
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Lecture 3: Solving First-order Linear ODE's; Steady-state and Transient Solution...
Jan 19, 2009
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Lecture 4: First-order Substitution Methods: Bernouilli and Homogeneous ODE's
Jan 19, 2009
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Lecture 5: First-order Autonomous ODE's: Qualitative Methods, Applications
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Lecture 6: Complex Numbers and Complex Exponentials
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Lecture 7: First-order Linear with Constant Coefficients: Behavior of Solutions,...
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Lecture 8: Continuation; Applications to Temperature, Mixing, RC-circuit, Decay...
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Lecture 9: Solving Second-order Linear ODE's with Constant Coefficients: The Thr...
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Lecture 10: Continuation: Complex Characteristic Roots; Undamped and Damped Osci...
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Lecture 11: Theory of General Second-order Linear Homogeneous ODE's: Superpositi...
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Lecture 12: Continuation: General Theory for Inhomogeneous ODE's. Stability Crit...
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Lecture 13: Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Fo...
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Lecture 14: Interpretation of the Exceptional Case: Resonance
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Lecture 15: Introduction to Fourier Series; Basic Formulas for Period 2(pi)
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Lecture 16: Continuation: More General Periods; Even and Odd Functions; Periodic...
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Lecture 17: Finding Particular Solutions via Fourier Series; Resonant Terms; Hea...
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Lecture 19: Introduction to the Laplace Transform; Basic Formulas
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Lecture 20: Derivative Formulas; Using the Laplace Transform to Solve Linear ODE...
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Lecture 21: Convolution Formula: Proof, Connection with Laplace Transform, Appli...
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Lecture 22: Using Laplace Transform to Solve ODE's with Discontinuous Inputs
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Lecture 23: Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer F...
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Lecture 24: Introduction to First-order Systems of ODE's; Solution by Eliminatio...
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Lecture 25: Homogeneous Linear Systems with Constant Coefficients: Solution via ...
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Lecture 26: Continuation: Repeated Real Eigenvalues, Complex Eigenvalues
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Lecture 27: Sketching Solutions of 2x2 Homogeneous Linear System with Constant C...
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Lecture 28: Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix...
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Lecture 29: Matrix Exponentials; Application to Solving Systems
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Lecture 30: Decoupling Linear Systems with Constant Coefficients
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Lecture 31: Non-linear Autonomous Systems: Finding the Critical Points and Sketc...
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Lecture 32: Limit Cycles: Existence and Non-existence Criteria
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Lecture 33: Relation Between Non-linear Systems and First-order ODE's; Structura...
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Unit I: First Order Differential Equations
Separable Equations
Mar 05, 2013
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Direction Fields
Mar 05, 2013
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Euler's Method
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Linear Equations
Mar 05, 2013
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Solutions of First Order Linear Equations
Mar 05, 2013
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Complex Numbers and Euler's Formula
Mar 05, 2013
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Sinusoidal Functions
Mar 05, 2013
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First Order Constant Coefficient Linear ODE's
Mar 05, 2013
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Sinusoidal Inputs
Mar 05, 2013
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Autonomous Equations and Phase Lines
Mar 05, 2013
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2408 Views
Unit II: Second Order Constant Coefficient Linear Equations
Homogeneous Constant Coefficient Equations: Real Roots
Mar 05, 2013
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2534 Views
Homogeneous Constant Coefficient Equations: Any Roots
Mar 05, 2013
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3077 Views
Forced Oscillations
Mar 05, 2013
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Gain and Phase Lag
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Undetermined Coefficients
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Pure Resonance
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Frequency Response
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Unit III: Fourier Series and Laplace Transform
Computing Fourier Series
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Manipulating Fourier Series
Mar 05, 2013
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2386 Views
Linear ODE's with Periodic Input
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Convolution and Green's Formula
Mar 05, 2013
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Partial Fractions and Laplace Inverse
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Laplace Transform: Basics
Mar 05, 2013
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Step and Delta Functions: Integration and Generalized Derivatives
Mar 05, 2013
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Unit Step and Impulse Response
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Pole Diagrams
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IV: First-order Systems
Linear Systems: Matrix Methods
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Matrix Exponentials
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Phase Portraits
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Linearization
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Damped Harmonic Oscillators
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Trace-Determinant Diagram
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Linear Systems: Complex Roots
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Linear Systems of Equations
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Laplace: Solving ODE's
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