Constructing a differential operator from its Green function

[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.

 

[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.

 

[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?...

 

[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.