MIT의 Differential Equations 강의 중 하나입니다.

2006년 봄에 강의한 것으로 고등학교 수학을 마친 사람이라면 큰 부담없이 들을 수 있다는 군요.

과연.. ㄱ=

http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/CourseHome/

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.


Lecture Note
SES # TOPICS
I. First-order Differential Equations
L0 Simple Models and Separable Equations
L1 Direction Fields, Existence and Uniqueness of Solutions (PDF)
L2 Numerical Methods (PDF)
L3 Linear Equations: Models (PDF)
L4 Solution of Linear Equations, Variation of Parameter (PDF)
L5 Complex Numbers, Complex Exponentials (PDF)
L6 Roots of Unity; Sinusoidal Functions (PDF)
L7 Linear System Response to Exponential and Sinusoidal Input; Gain, Phase Lag (PDF)
L8 Autonomous Equations; The Phase Line, Stability (PDF)

Muddy Card Responses (PDF)
L9 Linear vs. Nonlinear (PDF)
L10 Hour Exam I
II. Second-order Linear Equations
L11 The Spring-mass-dashpot Model; Superposition Characteristic Polynomial; Real Roots; Initial Conditions (PDF)

Muddy Card Responses (PDF)
L12 Complex Roots; Damping Conditions (PDF)
L13 Inhomogeneous Equations, Superposition (PDF)
L14 Operators and Exponential Signals (PDF)

Muddy Card Responses (PDF)
L15 Undetermined Coefficients (PDF)
L16 Frequency Response (PDF)
L17 Applications: Guest appearance by EECS Professor Jeff Lang (PDF)

Supplementary Notes

Driving Through the Dashpot (PDF)
L18 Exponential Shift Law; Resonance (PDF)
L19 Hour Exam II
III. Fourier Series
L20 Fourier Series (PDF)
L21 Operations on Fourier series (PDF)

Muddy Card Responses (PDF)
L22 Periodic Solutions; Resonance (PDF)
IV. The Laplace Transform
L23 Step Function and delta Function (PDF)
L24 Step Response, Impulse Response (PDF)
L25 Convolution (PDF)
L26 Laplace Transform: Basic Properties (PDF)

Muddy Card Responses (PDF)
L27 Application to ODEs; Partial Fractions (PDF)
L28 Completing the Square; Time Translated Functions (PDF)

Muddy Card Responses (PDF)
L29 Pole Diagram (PDF)
L30 Hour Exam III
V. First Order Systems
L31 Linear Systems and Matrices (PDF)
L32 Eigenvalues, Eigenvectors (PDF)
L33 Complex or Repeated Eigenvalues (PDF)
L34 Qualitative Behavior of Linear Systems; Phase Plane (PDF)
L35 Normal Modes and the Matrix Exponential (PDF)
L36 Inhomogeneous Equations: Variation of Parameters Again (PDF)

Muddy Card Responses (PDF)
L37 Nonlinear Systems (PDF)
L38 Examples of Nonlinear Systems (PDF)
L39 Chaos (PDF)
L40 Final Exam


Video Lecture
LEC # TOPICS STREAMING VIDEOs DOWNLOADABLE VIDEOs
1 The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 109MB)
2 Euler's Numerical Method for y'=f(x,y) and its Generalizations. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 126MB)
3 Solving First-order Linear ODE's; Steady-state and Transient Solutions. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 114MB)
4 First-order Substitution Methods: Bernouilli and Homogeneous ODE's. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 114MB)
5 First-order Autonomous ODE's: Qualitative Methods, Applications. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 103MB)
6 Complex Numbers and Complex Exponentials. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 100MB)
7 First-order Linear with Constant Coefficients: Behavior of Solutions, Use of Complex Methods. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 95MB)
8 Continuation; Applications to Temperature, Mixing, RC-circuit, Decay, and Growth Models. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG -117MB)
9 Solving Second-order Linear ODE's with Constant Coefficients: The Three Cases. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 114MB)
10 Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 103MB)
11 Theory of General Second-order Linear Homogeneous ODE's: Superposition, Uniqueness, Wronskians. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 111MB)
12 Continuation: General Theory for Inhomogeneous ODE's. Stability Criteria for the Constant-coefficient ODE's. (RM - 56K) (RM - 80K) (RM - 220K)
(MPEG - 103MB)
13 Finding Particular Sto Inhomogeneous ODE's: Operator and Solution Formulas Involving Ixponentials. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 98MB)
14 Interpretation of the Exceptional Case: Resonance. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 107MB)
15 Introduction to Fourier Series; Basic Formulas for Period 2(pi). (RM - 56K) (RM - 80K) (RM - 220K)
(MPEG - 118MB)
16 Continuation: More General Periods; Even and Odd Functions; Periodic Extension. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 105MB)
17 Finding Particular Solutions via Fourier Series; Resonant Terms; Hearing Musical Sounds. (RM - 56K) (RM - 80K) (RM - 220K)
(MPEG - 99MB)
19 Introduction to the Laplace Transform; Basic Formulas. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 95MB)
20 Derivative Formulas; Using the Laplace Transform to Solve Linear ODE's. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 103MB)
21 Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 83MB)
22 Using Laplace Transform to Solve ODE's with Discontinuous Inputs. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 83MB)
23 Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 93MB)
24 Introduction to First-order Systems of ODE's; Solution by Elimination, Geometric Interpretation of a System. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 90MB)
25 Homogeneous Linear Systems with Constant Coefficients: Solution via Matrix Eigenvalues (Real and Distinct Case). (RM - 56K) (RM - 80K) (RM - 220K)
(MPEG - 96MB)
26 Continuation: Repeated Real Eigenvalues, Complex Eigenvalues. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 97MB)
27 Sketching Solutions of 2x2 Homogeneous Linear System with Constant Coefficients. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 110MB)
28 Matrix Methods for Inhomogeneous Systems: Theory, Fundamental Matrix, Variation of Parameters. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 105MB)
29 Matrix Exponentials; Application to Solving Systems.
 (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 97MB)
30 Decoupling Linear Systems with Constant Coefficients. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 112MB)
31 Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 95MB)
32 Limit Cycles: Existence and Non-existence Criteria. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 95MB)
33 Relation Between Non-linear Systems and First-order ODE's; Structural Stability of a System, Borderline Sketching Cases; Illustrations Using Volterra's Equation and Principle. (RM - 56K) (RM - 80K) (RM - 220K) (MPEG - 96MB)
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