# Definition:Initial Condition

## Definition

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:

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

## Sources

- 1972: George F. Simmons:
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