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 satisfy.

That is, an initial condition is the additional imposition that a solution $y = y \left({x}\right)$ of $\Phi$ satisfy:


 * $y \left({x_0}\right) = y_0$

Also defined as
Some sources allow the initial condition to be an ordered $n$-tuple $\left({x_0, y_0, y_0', \ldots}\right)$, although this usage is relatively uncommon.

The imposition then becomes that, for all $i$ with $0 \le i \le n$, a solution $y$ satisfy:


 * $y^{\left({i}\right)} \left({x_0}\right) = y_0^{\left({i}\right)}$