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麻省理工開放課程:微分方程 Differential Equations 英文版 DVD 只於電腦播放



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).

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.