Recently bombed a quiz on Sturm-Liouville theory and orthogonal polynomials in my math methods for physics class, and I'm trying to go through the chapter on the theory and plug up the holes in the derivations the book assumes you to verify. I've been stuck on one for a while now.
The book says that the lefthand side of the following equation should be equal to zero: $$ \int _{a}^b y_{m}^*(x)Ly_{n}(x)\, dx-\int_{a}^b y_{n}(x)Ly_{m}^*(x) \, dx=(\lambda_{n}-\lambda_{m}^*)\int _{a}^b y_{n}(x)r(x)y^*(x)\, dx $$ and that we can prove that it will equal zero by getting that $$ \int _{a}^b y_{m}^*(x)Ly_{n}(x)\, dx-\int_{a}^b y_{n}(x)Ly_{m}^*(x) \, dx=\left[p(x)\{y_{m}^{*_{'}}(x)y_{n}(x)-y_{m}^*(x)y_{n}'(x)\}\right]_{a}^b=0 $$ For context $Ly_{n}(x)=\lambda_{n}r(x)y(x) $ and $Ly_{m}^*(x)=\lambda_{m}^*r(x)y^*(x)$.
I've gotten this far on the derivation of the lefthand side: $$ \int _{a}^b y_{m}^*(x)Ly_{n}(x)\, dx-\int_{a}^b y_{n}(x)Ly_{m}^*(x) \, dx $$ $$ =\int _{a}^b y_{m}^*(x)(-[p(x)y_{n}'(x)]'-q(x)y_{n}(x))\, dx-\int_{a}^b y_{n}(x)(-[p(x)y_{m}^{*_{'}}(x)]'-q(x)y_{m}^*(x)) \, dx $$ $$ \small =\int _{a}^b (-y_{m}^*(x)[p(x)y_{n}'(x)]'-q(x)y_{m}^*(x)y_{n}(x))\, dx-\int_{a}^b (-y_{n}(x)[p(x)y_{m}^{*_{'}}(x)]'-q(x)y_{n}(x)y_{m}^*(x)) \, dx $$ $$ \small =-\int _{a}^b y_{m}^*(x)[p(x)y_{n}'(x)]'\, dx-\int _{a}^bq(x)y_{m}^*(x)y_{n}(x))\, dx+\int_{a}^b (y_{n}(x)[p(x)y_{m}^{*_{'}}(x)]'\, dx+\int _{a}^bq(x)y_{n}(x)y_{m}^*(x)) \, dx $$ $$ =-\int _{a}^b y_{m}^*(x)[p(x)y_{n}'(x)]'\, dx+\int_{a}^b (y_{n}(x)[p(x)y_{m}^{*_{'}}(x)]'\, dx $$
$$ =-\int _{a}^b (y_{m}^*(x)(p'(x)y_{n}'(x)+p(x)y_{n}''(x)))\, dx+\int_{a}^b (y_{n}(x)(p'(x)y_{m}^{*_{'}}(x)+p(x)y_{m}^{*_{''}}(x)))\, dx $$ $$ \small =-\int _{a}^b y_{m}^*(x)p'(x)y_{n}'(x)\, dx -\int _{a}^b p(x)y_{m}^*(x)y_{n}''(x)\, dx+\int_{a}^b (y_{n}(x)p'(x)y_{m}^{*_{'}}(x)\, dx+\int_{a}^bp(x)y_{n}(x)y_{m}^{*_{''}}(x)\, dx $$ and then I don't know what to do from there. It seems from the book-provided answer, the second and fourth term should cancel or go to zero in some way, but I don't know how they would do so.
Further, how would $\left[p(x)\{y_{m}^{*_{'}}(x)y_{n}(x)-y_{m}^*(x)y_{n}'(x)\}\right]_{a}^b=0$? This chapter has me so lost but I know all of this leads into Green's functions, which I know are very useful, so I just wanna make sure I have the proper understanding of what comes before.