# Green,s inverse problem

1. Feb 16, 2005

### eljose

we all know that given a differential equation Ly=a0(x)y+a1(x)Dy+a2(x)D^2y=0 with Dy=dy/dx we can construct a Green function so LG(x,s)=d(x-s) being d the delta function.

My question is this given the G(x,s) could we construct the L so LG(x,s)=d(x-s) could someone provide an example of how to obtain?..thanks.

2. Feb 16, 2005

### dextercioby

HINT:Fourier transformed of the Green function coincides with the inverse of the Fourier transformed of the LINEAR differential operator...

Daniel.

P.S.Think aboout Poisson's equation for an infinite domain.

3. Feb 17, 2005

### eljose

You mean F[G(x,s)]=1/F[L(y)] so then F^-1[1/G(x,s)]=L(y) does the same happen with the Laplace transform?....

4. Feb 17, 2005

### dextercioby

I've never used Green functions and Laplace transforms...I told you what i knew...Maybe someone else could enlighten you more.

What i do know is that Cauchy problems in LINEAR ode's are dealt nicely by Laplace transform...But i don't recall some Green functions having been used...

Daniel.