Why are differential equations fundamental to science?
Science strives for some fundamental understanding of the world around us, and the universe as a whole (see for instance TOE). In an attempt to understand the motion of the planets, people such as Johannes Kepler described this motion in fairly precise mathematical terms, developing equations describing the path of motion that a planet will take. This motion was described in terms of ellipses. This sort of description is unsatisfactory since it gives no means of predicting the motion of more than two planets, or cannons, or bowling balls etc… When Isaac Newton wrote his Principia this changed. Newton’s insight was that while there are no fundamental laws describing the path of motion (is it a circle, an ellipse, or what? Imagine trying to come up with a fundamental law describing what the path taken by a tennis ball would be…) there are fundamental laws describing how that motion will change. Mathematically changes in motion are phrased in terms of derivatives, and therefore we are led
