\begin{definition}[Definition:Initial Condition]
Let $\Phi = \map F {x, y, y', y'', \ldots, y^{\paren n} }$ be an ordinary differential equation.
An '''initial condition''' is an ordered pair $\tuple {x_0, y_0}$ which any solution of $\Phi$ must satisfy.
That is, an '''initial condition''' is the additional imposition that a solution $y = \map y x$ of $\Phi$ satisfy:
:$\map y {x_0} = y_0$
\end{definition}