# Definition:Initial Condition

Jump to navigation
Jump to search

## Definition

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:

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

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 1.2$: General Remarks on Solutions