Differential Equations, Spring 2006 MIT

 Education

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 firstorder 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 Nonlinear autonomous systems: critical point analysis and phase plane diagrams.

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Lecture 01: The geometrical view of y'=f(x,y): direction fields, integral curves

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Lecture 02: Euler's numerical method for y'=f(x,y) and its generalizations

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Lecture 03: Solving firstorder linear ODE's; steadystate and transient solutions

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Lecture 04: Firstorder substitution methods: Bernouilli and homogeneous ODE's

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Lecture 05: Firstorder autonomous ODE's: qualitative methods, applications

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Lecture 06: Complex numbers and complex exponentials