# Sturm-Louville problem: u'' - u = x, u(0)=u(π)=0.

1. Apr 30, 2012

### Jamin2112

1. The problem statement, all variables and given/known data

2. Relevant equations

From the lecture notes:

"To study the Sturm-Louiville operator Ls in greater detail, we first need to determine it's adjoint operator, denoted by Ls. [...] The adjoint of the Sturm-Louiville operator satisfies the relationship

<Lsy1,y2>=<y1,Lsy2>,​

where y1 and y2 are two arbitrary functions satisfying the prescribed boundary conditions."

3. The attempt at a solution

I went straight ahead and tried to find the adjoint of the Sturm-Louiville operator. It got really messy, and I'm wondering whether there's something obvious that I'm missing.

2. May 3, 2012

### sunjin09

First, the operator Lu=u''-u≠x in general, otherwise there's no need to solve the DE.
Second, the boundary condition u(0)=u(pi)=0 applies to all u, v, w, etc. By letting <Lu,v>=<u,Ad{L}v>, and using integration by parts, it can be shown Ad{L}=L.

3. May 4, 2012

### Jamin2112

Got it.

Now the next part says

Well, since L*v = Lv = v '' - v, we need to solve v'' - v = 0 with v(0) = v(pi) = 0. Isn't the only solution v(x) = 0, since v '' - v = 0 implies v = C1ex + C2e-x????/