= Ordinary differential equation
{title2=$F(t,y,y',\ldots,y^{(n)})=0$}
{wiki}
= ODE
{c}
{synonym}
An <ordinary differential equation> relates an unknown <function> of one independent variable to its ordinary <derivatives>. Its order is the highest <derivative> appearing. Thus $y'=-k(y-T_a)$ is first order, while the <harmonic oscillator equation> $y''+\omega^2y=0$ is second order. A first-order equation in explicit form is
$$
y'(t)=f(t,y(t)).
$$
Here $t$ need not be time, and $y$ can be a <vector>, giving a system of coupled <ordinary differential equations>. A classical <solution of a differential equation> is a sufficiently <differentiable function> on an <interval> which satisfies the equation at every point. For the explicit first-order equation, if $f$ is continuous, this <solution of a differential equation> has a continuous <derivative>.
For <Newton's law of cooling>, the state is a single <temperature>; for a <second-order differential equation>, a first-order system also records its first <derivative>. This is why a starting position alone does not specify a <harmonic oscillator> trajectory: its starting velocity matters too.
The power of an <ordinary differential equation> is that one rule describes a whole family of histories. An <initial value problem> chooses one history when uniqueness holds. An <autonomous differential equation> has $f$ independent of $t$; its <equilibrium points> satisfy $f(y_*)=0$ and give constant <solutions of a differential equation>. Equations with explicit time dependence can instead respond to a changing environment.
Back to article page
