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18.03 Differential Equations

18.03 Differential Equations (Spring 2010, MIT OCW). Taught by Professor Arthur Mattuck, this course is a study of Ordinary Differential Equations (ODEs), including modeling physical systems. 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 (ODEs) deal with functions of one variable, which can often be thought of as time. Topics covered in this course include: Solution of first-order ODEs by analytical, graphical and numerical methods; Linear ODEs; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems; and Nonlinear autonomous systems. (from ocw.mit.edu)

Lecture 20 - Derivative Formulas

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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 ODEs; Steady-state and Transient Solutions

Lecture 04 - First-order Substitution Methods: Bernoulli and Homogeneous ODEs

Lecture 05 - First-order Autonomous ODEs: Qualitative Methods, Applications

Lecture 06 - Complex Numbers and Complex Exponentials

Lecture 07 - First-order Linear with Constant Coefficients

Lecture 08 - Continuation; Applications to Temperature, RC-circuit, Growth Models

Lecture 09 - Solving Second-order Linear ODEs with Constant Coefficients

Lecture 10 - Continuation: Complex Characteristic Roots; Undamped and Damped Oscillations

Lecture 11 - Theory of General Second-order Linear Homogeneous ODEs

Lecture 12 - Continuation: General Theory for Inhomogeneous ODEs

Lecture 13 - Finding Particular Solutions to Inhomogeneous ODEs

Lecture 14 - Interpretation of the Exceptional Case: Resonance

Lecture 15 - Introduction to Fourier Series

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 18

Lecture 19 - Introduction to the Laplace Transform

Lecture 20 - Derivative Formulas

Lecture 21 - Convolution Formula

Lecture 22 - Using Laplace Transform to Solve ODEs with Discontinuous Inputs

Lecture 23 - Use with Impulse Inputs; Dirac Delta Function, Weight and Transfer Functions

Lecture 24 - Introduction to First-order Systems of ODEs

Lecture 25 - Homogeneous Linear Systems with Constant Coefficients

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

Lecture 29 - Matrix Exponentials; Application to Solving Systems

Lecture 30 - Decoupling Linear Systems with Constant Coefficients

Lecture 31 - Non-linear Autonomous Systems

Lecture 32 - Limit Cycles: Existence and Nonexistence Criteria

Lecture 33 - Relation Between Non-linear Systems and First-order ODEs