Definition:Initial Condition

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Let $\Phi = F \left({x, y, y', y'', \ldots, y^{\left({n}\right)}}\right)$ 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:

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