Definition:Initial Condition

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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:

$\map y {x_0} = y_0$

Also defined as

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

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

$\map {y^{\paren i} } {x_0} = y_0^{\paren i}$