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【教学视频】 【麻省理工大学】微分方程(Differential Equations)
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转贴自：博普学习在线-->学习视频资料下载基地  Linear Phase Portraits Mathlet from the d'Arbeloff Interactive Math Project. (Image courtesy of Hu Hohn and Prof. Haynes Miller.) 课程重点 This course includes lecture notes, assignments, problems for group work in recitation, and a full set of lecture videos. 课程描述 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. 相关阅读资料  Listed in the table below are reading assignments for each lecture session. "EP" refers to the course textbook: Edwards, C. Henry, and David E. Penney. Elementary Differential Equations with Boundary Value Problems. 4th ed. "SN" refers to the "18.03 Supplementary Notes" written by Prof. Miller. "Notes" refers to the "18.03 Notes and Exercises" written by Prof. Mattuck. 课 课程单元 阅读资料 I. First-order Differential Equations 1 Introduction Separable Equations Direction Fields EP 1.3 Notes G.1 (PDF) SN §1 (PDF) 2 Isoclines Models EP 1.3, 1.4 SN §2 (PDF) 3 Linear Equations EP 1.5 SN §3 (PDF) 4 Autonomous Equations The Phase Line EP 1.7, 7.1 5 Complex Numbers Complex Exponential SN §5 (PDF) SN §6 (PDF) Notes C.1–3 (PDF) 6 Sinusoidal Functions SN §4 (PDF) Notes IR.6 (PDF) 7 Sinusoidal System Response Notes IR.5 (PDF) 8 Hour Exam I II. Second-order Linear Equations 9 Solutions of Spring-mass-dashpot Models EP 2.1, 2.3 10 Superposition Initial Conditions EP 2.2 SN §9 (PDF) 11 Damping Conditions Inhomogeneous Equations For Damping Conditions EP 2.4 For Inhomogeneous Equations Notes O.1 (PDF) EP 2.6 (pp. 158–159 only; see SN §7 (PDF) if you want to learn about beats) 12 Exponential Signals SN §10 (PDF) EP 2.6 (pp. 165–167) 13 Operator Notation and Undetermined Coefficients SN §11 (PDF) EP 2.5 (pp. 144–153) Notes O.1, 2, and 4 (PDF) 14 Frequency Response SN §13 (PDF) SN §14 (PDF) 15 Resonance SN §12 (PDF) Notes O.3 (PDF) 16 Review 17 Hour Exam II III. Delta Functions and Convolution 18 Step and Delta Functions SN §16 (PDF) 19 Impulse Response and Convolution SN §17 (PDF) Notes I (PDF) 20 From Convolution to the Laplace Transform IV. The Laplace Transform 21 Laplace Transform: Basic Properties EP 4.1 22 Application to ODEs Partial Fractions SN §18 (PDF) EP 4.2, 4.3 23 Completing the Square Transforms of Delta and Time Translated Functions EP 4.5–4.6 24 Convolution and Laplace Transform The Pole Diagram EP 4.4 SN §19 (PDF) 25 Numerical Methods EP 6.1 Notes G (PDF) V. Fourier Series 26 Fourier Series EP 8.1 27 Differentiating and Integrating EP 8.3 28 General Period EP 8.2 29 Periodic Solutions EP 8.3, 8.4 30 Review: Fourier, Euler, Laplace 31 Hour Exam III VI. First-order Systems 32 Linear Systems and Matrices EP 5.1–5.3 SN §23 (PDF) Notes LS.1 (PDF) 33 Eigenvalues Eigenvectors EP 5.4 Notes LS.2 (PDF) 34 Complex or Repeated Eigenvalues EP 5.4 Notes LS.3 (PDF) 35 Qualitative Behavior of Linear Systems SN §24 (PDF) 36 Normal Modes and the Matrix Exponential EP 5.7 Notes LS.6 (PDF) 37 Inhomogeneous Equations EP 5.8 38 Nonlinear Systems The Phase Plane EP 7.2, 7.3 Notes GS (PDF) 39 Examples of Nonlinear Systems EP 7.4, 7.5 Notes GS (PDF) 40 Final Exam 影片教学 本课程正在进行字幕听打计划，如需观看成果或协助，请浏览相关网页。 RealOne™ Player software is required to run the .rm files in this section. These video lectures of Professor Arthur Mattuck teaching 18.03 were recorded live in the Spring 2003 and do not correspond precisely to the lectures taught in the Spring of 2004. Professor Mattuck has inspired and informed generations of MIT students with his engaging lectures. The videotaping was made possible by The d'Arbeloff Fund for Excellence in MIT Education . Note: Lecture 18, 34, and 35 are not available. Lecture #1: The geometrical view of y'=f(x,y): direction fields, integral curves. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #17: Finding particular solutions via Fourier series; resonant terms;hearing musical sounds. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #2: Euler's numerical method for y'=f(x,y) and its generalizations. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #19: Introduction to the Laplace transform; basic formulas. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #3: Solving first-order linear ODE's; steady-state and transient solutions. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #20: Derivative formulas; using the Laplace transform to solve linear ODE's. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #4: First-order substitution methods: Bernouilli and homogeneous ODE's. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #21: Convolution formula: proof, connection with Laplace transform, application to physical problems. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #5: First-order autonomous ODE's: qualitative methods, applications. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #22: Using Laplace transform to solve ODE's with discontinuous inputs. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #6: Complex numbers and complex exponentials. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #23: Use with impulse inputs; Dirac delta function, weight and transfer functions. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #7: First-order linear with constant coefficients: behavior of solutions, use of complex methods. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #24: Introduction to first-order systems of ODE's; solution by elimination, geometric interpretation of a system. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #8: Continuation; applications to temperature, mixing, RC-circuit, decay, and growth models. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #25: Homogeneous linear systems with constant coefficients: solution via matrix eigenvalues (real and distinct case). (RM - 56K) (RM - 80K) (RM - 220K) Lecture #9: Solving second-order linear ODE's with constant coefficients: the three cases. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #26: Continuation: repeated real eigenvalues, complex eigenvalues. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #10: Continuation: complex characteristic roots; undamped and damped oscillations. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #27: Sketching solutions of 2x2 homogeneous linear system with constant coefficients. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #11: Theory of general second-order linear homogeneous ODE's: superposition, uniqueness, Wronskians. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #28: Matrix methods for inhomogeneous systems: theory, fundamental matrix, variation of parameters. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #12: Continuation: general theory for inhomogeneous ODE's. Stability criteria for the constant-coefficient ODE's. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #29: Matrix exponentials; application to solving systems. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #13: Finding particular solutions to inhomogeneous ODE's: operator and solution formulas involving exponentials. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #30: Decoupling linear systems with constant coefficients. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #14: Interpretation of the exceptional case: resonance. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #31: Non-linear autonomous systems: finding the critical points and sketching trajectories; the non-linear pendulum. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #15: Introduction to Fourier series; basic formulas for period 2(pi). (RM - 56K) (RM - 80K) (RM - 220K) Lecture #32: Limit cycles: existence and non-existence criteria. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #16: Continuation: more general periods; even and odd functions; periodic extension. (RM - 56K) (RM - 80K) (RM - 220K) Lecture #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) |
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