An Introduction To - Differential Equations: With...

“But the universe doesn’t sit still for portraits. The universe is a movie. And if you want to understand the movie, you don't look at the frames; you look at the between them.” He drew a single, elegant equation: dy/dx = ky .

As Elias spoke, the chalkboard filled with the language of the shifting world: , where one side of the world is pulled away from the other to find clarity; Integrating Factors , the "magic" multipliers that turn chaos into a perfect derivative; and Initial Conditions , the single "X marks the spot" that tells you which of a thousand possible paths the universe actually took. An Introduction to Differential Equations: With...

“To solve a standard equation is to find a hidden number. But to solve a differential equation is to find a . You aren't looking for a '7' or a '10.' You are looking for a function—a curve that describes the path of a planet or the vibration of a violin string.” “But the universe doesn’t sit still for portraits

He began to sketch a , a sea of tiny marks that looked like iron filings caught in a magnetic web. “We start with the rate. We start with the 'how fast.' And from that sliver of motion, we reconstruct the entire history of the system.” As Elias spoke, the chalkboard filled with the