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$$\partial_x^2y + \delta(x) \partial_x y + y= 0$$

where ##\delta(x)## is the dirac delta function. I need the solution for periodic boundary conditions from ##-\pi## to ##\pi##.

I've realised that I can solve this for some types of boundary conditions. What i'd be really interested in is how to do this for periodic boundary conditions...

Technically, if I approach the problem by splitting the regions ##x<0## and ##x>0## and solve in each part separately, I can solve it and get linear equations in both regions. This will give me ##4## variables. Periodicity, and periodicity of the derivative will give me 2 equations. Continuity at ##x=0## will give me one more. How do i relate the derivative around the ##x=0## interface?

I guess I should make my actual problem a bit clearer as well. I'm basically interested in some technique by which I can get the information for the change in derivative of the function around the delta function.

A little background: If there was no delta function, but rather say some gaussian approximation, I would be expect to be able to solve it, but I don't see why I can't get the information of the derivative around ##x=0## when i put in a dirac delta function. My actual problem is reasonably more complicated but this is the quickest simple example I could reduce my problem to. If I try to integrate in an epsilon region around ##0##, then I end up with an expression in ##y^\prime(0)##, which isn't defined.

Any help or direction would be greatly appreciated!