Newton already sorted differential equations into three kinds
A differential equation links unknown functions to their derivatives—positions to velocities, heats to fluxes. Newton's 1671 work on fluxions listed three types; Bernoulli's 1695 equation and Fourier's 1822 heat book show how far the tool then spread.
Study centers on solution sets and their properties; only the simplest equations admit tidy closed forms, so numerical approximation fills modern practice. In Newton's examples the unknown's derivative equals a function of x, or of x and y, or a first-order partial relation; he used infinite series and noted non-unique solutions.
Jacob Bernoulli posed his namesake ordinary equation in 1695; Leibniz simplified it the next year. Vibrating-string debates drew d'Alembert, Euler, Daniel Bernoulli, and Lagrange; the Euler–Lagrange equation emerged in the 1750s from tautochrone work. Fourier's Analytic Theory of Heat in 1822 built on Newton's cooling law to treat heat flow.
D'Alembert found the one-dimensional wave equation in 1746, and Euler reached the three-dimensional version within a decade. Lagrange cracked the tautochrone, the curve down which a weighted particle reaches a fixed point in the same time from any start, in 1755 and mailed his answer to Euler. In classical mechanics, Newton's laws turn position, velocity and forces into an equation of motion for where a body sits over time; a ball falling through air, for example, accelerates by gravity minus the drag of air resistance. Ordinary equations involve a function of one variable, while partial ones handle several variables and describe sound, heat, electrostatics, fluid flow, elasticity and quantum mechanics. Linear equations have a well-developed theory, covering cases such as radioactive decay and heat diffusion. Solutions of linear equations also serve to define many special functions.
Source: Differential equation
