# How to handle a differential equation whose RHS contains the derivative of the Dirac delta function?

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}

• Actually ,my the goal is to extract the solution using Green's function approach Mar 16, 2019 at 11:31
• Actually I was editing the question, It may be happened accddentally. Please suggest , so that I can make changes to the question Mar 16, 2019 at 11:33
• Assuming zero initial conditions (all four of them), the Laplace transform is $$\Psi (s) = \psi'(\zeta) \left(\frac{s}{s^4 - \beta^4}\right)$$ From here, one can use partial fraction expansion to conclude what the form of the solution is. Mar 16, 2019 at 11:47
• This is using Laplace transforms. Is there any ways that I can fit a Green's function to this DE. Mar 16, 2019 at 11:49
• I am not familiar with Green's functions. At least, not that I know of. Mar 16, 2019 at 11:50