I have a differential equation of the following form.
\begin{align} \begin{split} \frac{\mathrm{d}^4\psi(\eta)}{\mathrm{d}\eta^4}-\beta^4\psi(\eta)=\psi'(\zeta)\,\delta'(\eta-\zeta) \end{split} \end{align}
How to handle the derivative of the Dirac delta function on the right-hand side? I am trying to find a closed form solution using Green's functions.
Suppose if a differential equation contains both Dirac delta function and its derivative, can I linearly superimpose the solution of the differential equation of just Dirac delta function plus the solution of the differential equation with derivative of Dirac delta function?
\begin{align} \begin{split} \frac{\mathrm{d}^4\psi(\eta)}{\mathrm{d}\eta^4}-\beta^4\psi(\eta)=\psi(\zeta)\,\delta(\eta-\zeta)+\psi'(\zeta)\,\delta'(\eta-\zeta) \end{split} \end{align}