 Differential equation

A differential equation is an equation relating an unknown function to its derivatives. It specifies how change depends on the present state, and sometimes on the independent variables themselves. A solution of a differential equation is a function satisfying that relation throughout a stated domain; giving a formula without its domain can conceal a point where the equation is undefined.

The central idea is to turn a local law into a whole trajectory. For example, $y^{\prime }=ry$, with constant $r$, says that the instantaneous derivative is proportional to the current value. If $y(0)=100$ and $r=0.2$ per day, the solution of a differential equation is $y(t)=100e^{0.2t}$, so $y(5)=100e\approx 271.83$. This is an ideal growth model assuming a constant proportional rate; the exponential function emerges from repeatedly applying the same local rule. Changing the sign of $r$ gives exponential decay.

An ordinary differential equation uses one independent variable. A partial differential equation can involve several, such as space and time, and uses partial derivatives. Additional information selects a particular solution of a differential equation: an initial condition describes a starting state, whereas a boundary condition imposes constraints at the edge of a domain. The useful questions extend beyond finding a formula: does a solution of a differential equation exist, is it unique, and what happens to it over time?