# Solution of differential equation with Dirac Delta

1. Jul 8, 2014

### SarthakC

Is it possible to solve a differential equation of the following form?

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

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!

2. Jul 9, 2014

### the_wolfman

Have you tried using a Fourier transform?

3. Aug 23, 2014