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

[Jamin2112]

Homework Statement

Homework Equations

From the lecture notes:

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

<L_sy₁,y₂>=<y₁,L_s^†y₂>,​

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

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.

 

[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.

 

[Jamin2112]

Got it.

Now the next part says

Solve the associated homogeneous adjoint problem problem L*v = 0 with the same boundary conditions. Show that Lu = u'' - u = x has a unique solutions without actually solving it.

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 = C₁e^x + C₂e^-x?/