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## Created by: ## Last updated:

December 23, 2022

## Duration:

Unlimited Duration

FREE

## This course includes:

Unlimited Duration

## Certificate of completion

Unlimited Duration

### Description

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.

### Course Curriculum

• Direction fields, existence and uniqueness of solutions Unlimited
• Numerical methods Unlimited
• Linear equations, models Unlimited
• Solution of linear equations, integrating factors Unlimited
• Complex numbers, roots of unity Unlimited
• Complex exponentials; sinusoidal functions Unlimited
• Linear system response to exponential and sinusoidal input; gain, phase lag Unlimited
• Autonomous equations; the phase line, stability Unlimited
• Linear vs. nonlinear Unlimited
• Modes and the characteristic polynomial Unlimited
• Good vibrations, damping conditions Unlimited
• Exponential response formula, spring drive Unlimited
• Complex gain, dashpot drive Unlimited
• Operators, undetermined coefficients, resonance Unlimited
• Frequency response Unlimited
• LTI systems, superposition, RLC circuits Unlimited
• Engineering applications Unlimited
• Fourier series Unlimited
• Operations on fourier series Unlimited
• Periodic solutions; resonance Unlimited
• Step functions and delta functions Unlimited
• Step response, impulse response Unlimited
• Convolution Unlimited
• Laplace transform: basic properties Unlimited
• Application to ODEs Unlimited
• Second order equations; completing the squares Unlimited
• The pole diagram Unlimited
• The transfer function and frequency response Unlimited
• Linear systems and matrices Unlimited
• Eigenvalues, eigenvectors Unlimited
• Complex or repeated eigenvalues Unlimited
• Qualitative behavior of linear systems; phase plane Unlimited
• Normal modes and the matrix exponential Unlimited
• Nonlinear systems Unlimited
• Linearization near equilibria; the nonlinear pendulum Unlimited
• Limitations of the linear: limit cycles and chaos Unlimited

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Students Massachusetts Institute of Technology