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1. Show that the operator:

L=-(1-x^2)d2/dx2+2xd/dx is self adjoint if both y(1) and y(-1) are required to be finite.

I've somehow done some integration by parts over an arbitrary interval and shown that the integral of f*(x)[Lg(x)] is equal to the integral of
[Lf(x)]*g(x) + some boundary terms. But I don't understand the requirement that y(1) and y(-1) must be finite for L to be self adjoint, looking at the operator I can see that x=+(-1) are going to be important points and applying the operator to y will give an ODE of the form: y''+p(x)y'+q(x)y=0 which means p(x) and q(x) will be singular and non-differentiable at x=1 and -1, but I don't see how to tie this in with the requirement above.

Cheers.
2. For the operator to be self adjoint you require <Ly,w>=<y,Lw> so you need the boundary terms to be zero using the fact that y is finite at +/-1
For the operator to be self adjoint you require <Ly,w>=<y,Lw> so you need the boundary terms to be zero using the fact that y is finite at +/-1
I'm pretty sure this is incorrect. If the boundary terms sum to zero this indicates that the operator is Hermitian as well as being self adjoint.
4. Hermitian and self adjoint are the same thing. The operator incudes the boundary conditions aswell
5. Ok, the last part of this question asks me to find the solution of the equation Ly=x^3 in terms of p1 and p3, where:

p1=x and p3=1/2(5x^3-3x) are the first 2 odd Legendre polynomials.

I've expressed x^3 in terms of the eigenfunction expansion of the Legendre polynomials above if thats necessary. But how do I solve this inhomogenous ODE and get a solution in terms of p1 and p3? Cheers.
6. After thinking more about this, I think I must find the solution to the ODE and then just express it in terms of the p1 and p2 using an eigenfunction expansion?

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