Definition:Initial Condition

Definition
Let $\Phi = F \left({x, y, y\,^{\prime}, y\,^{\prime \prime}, \ldots, y^{\left({n}\right)}}\right)$ be an ordinary differential equation.

An initial condition is an ordered pair $\left({x_0, y_0}\right)$ which any solution of $\Phi$ must satisy.

Some sources allow the initial condition to be an ordered n-Tuple $\left({x_0, y_0, y\,^{\prime}\ldots}\right)$, although this usage is relatively uncommon.