Definition:Initial Condition

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:

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

Also see

• Results about initial conditions can be found here.

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): boundary conditions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): initial conditions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): boundary conditions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): initial conditions
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): initial value