Ordinary differential equation

An ordinary differential equation (ODE) relates an unknown function $y(t)$ of one independent variable to its derivatives. A first-order system in explicit form is

y'(t)=F(t,y(t)),\qquad y(t)\in\mathbb R^m

or $\mathbb C^m$ . A classical solution is a differentiable function on an interval that satisfies the equation at every point. Higher-order equations can often be written as first-order systems by including successive derivatives as additional state variables.

Data and parameters

An initial value selects the state at one time. Additional parameters may enter $F$ without being differentiated in $t$ . An equation is autonomous when $F$ does not depend explicitly on $t$ . Ordinary refers to the single independent variable, even when the state has many components.