Solving Laplacian Operators and DEs

[Hertz]

Hi, lately I've been messing around a lot with the Laplacian operator and DE's including the Laplacian operator. Most recently, the equation below is the one I have been messing around with and trying to understand better.

$\nabla^2 U(\vec{r})=C(\vec{r})U(\vec{r})$

This is pretty general though.. WAYY too general for me to tackle. So I've been starting with the 1D case, which I also can't seem to solve.

$\frac{d^2}{dx^2}U(x)=C(x)U(x)$

My goal is to try to solve for U(x) in terms of C(x). Any ideas? Is there any way to know if such a solution exists? What about to the general equation above?

Thanks :)

 

[HallsofIvy]

Do you know the form of the Laplacian operator in spherical or polar coordinates

 

[Hertz]

No I don't but it wouldn't be too much of a hassle to figure it out. How could that help though?

 

[the_wolfman]

A couple of thoughts for progressing.

1) Try the case where C is constant. This actually gives you a Helmholtz equation.

2) For the more general case, it helps to assume that U or U dot n =0 at the boundary and C has a certain sign.
Then multiply by U and integrate over the domain, this will involve an integration by parts.