= Differential equation
{wiki}
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, $y'=ry$, 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 $y(5)=100e\approx271.83$. 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?
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