Differential equation

An equation relating a function to its own derivative(s), like $y'=ky$ .

A differential equation relates a function to one or more of its own derivatives, like $y'=ky$ or $y''+4y=0$ . Solving one means finding the function(s) $y(x)$ that make the equation true — not a number, but an entire FUNCTION. The ORDER of a differential equation is the highest derivative appearing in it (first-order involves only $y'$ , second-order also $y''$ , and so on).

Examples

$y'=2x$ is first-order; its general solution is $y=x^2+C$ (any antiderivative works).
$y'=y$ is first-order; $y=Ce^x$ solves it for any constant $C$ , since $(Ce^x)'=Ce^x$ .