# Definition:Boundary Condition

## Definition

Let $\Phi$ be a differential equation to which a particular solution is to be found.

A boundary condition is an equation relating particular values of the dependent and independent variables which the particular solution to $\Phi$ must satisfy.

It is usual for such boundary conditions to correspond to the physical extremities of bodies, aspects of whose internal nature is being modelled by means of $\Phi$.

## Examples

### Arbitrary Example

Consider the differential equation:

$\dfrac {\d^2 y} {\d x^2} + 4 \dfrac {\d y} {\d x} = 0$

for $x \ge 0$.

From Solution to $y + 4 y' = 0$ this has a general solution:

$y = A + B e^{-4 x}$

Let the boundary conditions be:

 $\ds y$ $=$ $\ds 0$ when $x = 0$ $\ds \dfrac {\d y} {\d x}$ $=$ $\ds 1$ when $x = 0$

Then substituting $x = 0$ into the general solution and its first derivative yields:

 $\ds A$ $=$ $\ds \dfrac 1 4$ $\ds B$ $=$ $\ds -\dfrac 1 4$

and so the particular solution:

$4 y = 1 - e^{-4 x}$

## Also see

• Results about boundary conditions can be found here.