Is it possible to solve a differential equation of the following form?
$\partial_x^2y + \delta(x) \partial_x y = 0$
where $\delta(x)$ is the dirac delta function. I need the solution for periodic boundary conditions from -1 to 1,
but I'll be fine if you direct me for any sort of boundary conditions..
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?
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.
If it helps, i'm actually interested in a physical problem, so this delta function is just an approximation, but i'm unable to take any function in it's place whose limit I can make tend to a delta function..
Any help or direction would be greatly appreciated!
EDIT: I guess I should make my actual problem a bit clearer as well. In this simplistic case i can simplify the differential equation by means of the substitution of $z=y^\prime$, however I can't really do that in my original equation. 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 better example might be $\partial_x^2y + \delta(x) \partial_x y +y= 0$