麻省理工开放课程：微分方程 Differential Equations 英文字幕包/共33课更新完毕[MP4]

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资源信息： 中文名: 麻省理工开放课程：微分方程 英文名: Differential Equations 资源格式: MP4 课程类型: 数学 学校: 麻省理工 MIT 主讲人: Prof. Arthur Mattuck 版本: 英文字幕包/共33课更新完毕 发行日期: 2006年 地区: 美国 对白语言: 英语 概述:

课程介绍： 微分方程是一门表述自然法则的语言。理解微分方程解的性质，是许多当代科学和工程的基础。常微分方程（ODE's）是关于单变量的函数，一般可以认为是时域变量。学习内容包括：利用解释、图形和数值方法求解一阶常微分方程，线性常微分方程，尤指二阶常系数方程，不定系数和参变数，正弦和指数信号：振动、阻尼和共振，复数和幂，傅立叶级数，周期解，Delta函数、卷积和拉普拉斯变换方法，矩阵和一阶线性系统：特征值和特征向量，非线性独立系统：临界点分析和相平面图。 导师介绍 Arthur Mattuck is a tenured Professor of Mathematics at the Massachusetts Institute of Technology. He may be best known for his 1998 book, Introduction to Analysis (ISBN 013-0-81-1327) and his differential equations video lectures featured on MIT's OpenCourseWare. Inside the department, he is well known to graduate students and instructors, as he watches the videotapes of new recitation teachers (an MIT-wide program in which the department participates). 截图：

注： 1.我会把麻省所有的微积分课程发出来，大约200个视频，有需要的同学请关注。 2.课件：http://ocw.mit.edu/courses/mathematics/18-...terials/，因为官网速度下载较快，这里暂不提供。 3.个人翻译的标题,有啥错误还请慷慨指正. 目录: Lecture 01: The geometrical view of y'=f(x,y): direction fields, integral curves. Lecture 02: Euler's numerical method for y'=f(x,y) and its generalizations. Lecture 03: Solving first-order linear ODE's; steady-state and transient solutions. Lecture 04: First-order substitution methods: Bernouilli and homogeneous ODE's. Lecture 05: First-order autonomous ODE's: qualitative methods, applications. Lecture 06: Complex numbers and complex exponentials. Lecture 07: First-order linear with constant coefficients: behavior of solutions, use of complex methods. Lecture 08: Continuation; applications to temperature, mixing, RC-circuit, decay, and growth models. Lecture 09: Solving second-order linear ODE's with constant coefficients: the three cases. Lecture 10: Continuation: complex characteristic roots; undamped and damped oscillations. Lecture 11: Theory of general second-order linear homogeneous ODE's: superposition, uniqueness, Wronskians. Lecture 12: Continuation: general theory for inhomogeneous ODE's. Stability criteria for the constant-coefficient ODE's. Lecture 13: Finding particular solutions to inhomogeneous ODE's: operator and solution formulas involving exponentials. Lecture 14: Interpretation of the exceptional case: resonance. Lecture 15: Introduction to Fourier series; basic formulas for period 2(pi). Lecture 16: Continuation: more general periods; even and odd functions; periodic extension. Lecture 17: Finding particular solutions via Fourier series; resonant terms;hearing musical sounds. Lecture 19: Introduction to the Laplace transform; basic formulas. Lecture 20: Derivative formulas; using the Laplace transform to solve linear ODE's. Lecture 21: Convolution formula: proof, connection with Laplace transform, application to physical problems. Lecture 22: Using Laplace transform to solve ODE's with discontinuous inputs. Lecture 23: Use with impulse inputs; Dirac delta function, weight and transfer functions. Lecture 24: Introduction to first-order systems of ODE's; solution by elimination, geometric interpretation of a system. Lecture 25: Homogeneous linear systems with constant coefficients: solution via matrix eigenvalues (real and distinct case). Lecture 26: Continuation: repeated real eigenvalues, complex eigenvalues. Lecture 27: Sketching solutions of 2x2 homogeneous linear system with constant coefficients. Lecture 28: Matrix methods for inhomogeneous systems: theory, fundamental matrix, variation of parameters. Lecture 29: Matrix exponentials; application to solving systems. Lecture 30: Decoupling linear systems with constant coefficients. Lecture 31: Non-linear autonomous systems: finding the critical points and sketching trajectories; the non-linear pendulum. Lecture 32: Limit cycles: existence and non-existence criteria. 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.