Boundary Condition/Examples/Arbitrary Example 1

Example of Boundary Condition

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:

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

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): boundary conditions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): boundary conditions