Differential Equations
A tour through some of the most beautiful methods for studying dynamic systems
Differential equations, studying the unsolvableWhat is a differential equation, the pendulum equation, and some basic numerical methods
Solving the heat equationSolving the heat equation.
But what is a Fourier series? From heat flow to circle drawingsHow Fourier series arose from studying heat flow, and how they can be thought of as decomposing any drawing in 2d into a sum of rotations.
The Physics of Euler's Formula | Laplace Transform PreludeHow dynamics explain Euler's formula, and vice versa. Solutions to differential equations can often be broken down as sums of complex exponentials, and understanding this sets up an understanding of the Laplace Transform.
But what is a Laplace Transform?Visualizing the most important tool for differential equations, and how it decomposes functions into exponential pieces.
Why Laplace transforms are so usefulStudying the forced harmonic oscillator by taking a Laplace transform and studying its poles.
e^(iπ) in 3.14 minutes, using dynamicsA quick explanation of e^(pi i) in terms of motion and differential equations
How (and why) to raise e to the power of a matrixExponentiating matrices, and the kinds of linear differential equations this solves.
Exponential growth and epidemicsA primer on exponential and logistic growth, with epidemics as a central example
The diffusion equationHow the diffusion equation can arise from a simple random walk model. Week 12, MIT 18.S191 Fall 2020