Green function differential equation

Green’s function differential equation is typically used to solve linear inhomogeneous differential equations.

[latex]
\[
L G(x, s) = \delta(x – s)
\]

Here:

• 
  LL 

  $L$ is the differential operator.

• 
  G(x,s)G(x, s) 

  $G(x,s)$ is the Green’s function.

• 
  δ(x−s)\delta(x – s) 

  $\delta (x-s)$ is the Dirac delta function.

Example for a Specific Case:

For the operator

L=d2dx2−k2L = \frac{d^2}{dx^2} – k^2

L=dx2d2​−k2:

[latex]
\[
\left( \frac{d^2}{dx^2} – k^2 \right) G(x, s) = \delta(x – s)
\]