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:

• $L$  $L$ is the differential operator.
• $G(x, s)$  $G(x,s)$ is the Green’s function.
• $\delta(x – s)$  $\delta (x-s)$ is the Dirac delta function.

Example for a Specific Case:

For the operator $L = \frac{d^2}{dx^2} – k^2$ : [latex] \[ \left( \frac{d^2}{dx^2} – k^2 \right) G(x, s) = \delta(x – s) \]