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A Conversation with Brian Hayes, Writer and Photographer

Description: Felice Frankel and Brian Hayes have a conversation about what exactly goes on, from a technical perspective, when you make an image. Instructors: Felice Frankel and Brian Hayes

A Conversation with Christine Daniloff, Creative Director at MIT News

Description: Felice Frankel has a conversation with Christine Daniloff about the images used on the MIT homepage. Instructor: Felice Frankel and Christine Daniloff

A Conversation with J. Kim Vandiver, Director of the MIT Edgerton Center

Description: Felice Frankel has a conversation with J. Kim Vandiver about Doc Edgerton and the history of the Edgerton Center. Instructors: Felice Frankel and J. Kim Vandiver

MIT Learn Differential Equations

Gilbert and Cleve Introduction

Overview of Differential Equations

Description: Differential equations connect the slope of a graph to its height. Slope = height, slope = -height, slope = 2t times height: all linear. Slope = (height)2 is nonlinear. Related section in textbook: 1.1 Instructor: Prof. Gilbert Strang

The Calculus You Need

Description: The sum rule, product rule, and chain rule produce new derivatives from known derivatives. The Fundamental Theorem of Calculus says that the integral inverts the derivative. Related section in textbook: 1.2 Instructor: Prof. Gilbert Strang

Response to Exponential Input

Description: For a linear equation with exponential input from outside and exponential growth from inside, the solution is a combination of two exponentials. Related section in textbook: 1.4b Instructor: Prof. Gilbert Strang

Response to Oscillating Input

Description: A linear equation with oscillating input has an oscillating output with the same frequency (and a phase shift). Related section in textbook: 1.4c Instructor: Prof. Gilbert Strang

Solution for Any Input

Description: To solve a linear first order equation, multiply each input by its growth factor and integrate those outputs. Related section in textbook: 1.4d Instructor: Prof. Gilbert Strang

Step Function and Delta Function

Description: A unit step function jumps from 0 to 1. Its slope is a delta function: Zero everywhere except infinite at the jump. Related section in textbook: 1.4e Instructor: Prof. Gilbert Strang

Response to Complex Exponential

Description: For linear equations, the solution for a cosine input is the real part of the solution for a complex exponential input. That complex solution has magnitude G (the gain). Related section in textbook: 1.5 Instructor: Prof. Gilbert Strang

Integrating Factor for Constant Rate

Description: The integrating factor multiplies the differential equation to allow integration. Related section in textbook: 1.6 Instructor: Prof. Gilbert Strang

Integrating Factor for a Varying Rate

Description: The integral of a varying interest rate provides the exponent in the growing solution (the bank balance). Related section in textbook: 1.6b Instructor: Prof. Gilbert Strang

The Logistic Equation

Description: When competition slows down growth and makes the equation nonlinear, the solution approaches a steady state. Related section in textbook: 1.7 Instructor: Prof. Gilbert Strang

The Stability and Instability of Steady States

Description: Steady state solutions can be stable or unstable—a simple test decides. Related section in textbook: 1.7c Instructor: Prof. Gilbert Strang

Separable Equations

Description: Separable equations can be solved by two separate integrations, one in t and the other in y. Related section in textbook: 1.8 Instructor: Prof. Gilbert Strang

Second Order Equations

Description: For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency. Related section in textbook: 2.1 Instructor: Prof. Gilbert Strang

Forced Harmonic Motion

Description: When the forcing is a sinusoidal input, like a cosine, one particular solution has the same form. But if the forcing frequency equals the natural frequency there is resonance. Related section in textbook: 2.1b Instructor: Prof. Gilbert Strang

Unforced Damped Motion

Description: With constant coefficients in a differential equation, the basic solutions are exponentials. The exponent solves a simple equation. Related section in textbook: 2.3 Instructor: Prof. Gilbert Strang

Impulse Response and Step Response

Description: The impulse response is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition. Related section in textbook: 2.3c Instructor: Prof. Gilbert Strang

Exponential Response – Possible Resonance

Description: Resonance occurs when the natural frequency matches the forcing frequency—equal exponents from inside and outside. Related section in textbook: 2.4 Instructor: Prof. Gilbert Strang

Second Order Equations with Damping

Description: A damped forced equation has a sinusoidal solution with exponential decay. The damping ratio provides insight into the null solutions. Related section in textbook: 2.4b Instructor: Prof. Gilbert Strang

Electrical Networks: Voltages and Currents

Description: Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance). Related section in textbook: 2.5 Instructor: Prof. Gilbert Strang