Definition:Initial Condition

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.