For a homework problem, I'm supposed to prove Lagrange's identity using integration by parts. That is, I'm supposed to show that
$uSv-vSu=\frac{1}{w}\frac{d}{dx}[p(uv'-u'v)]$
Where $S$ is a Sturm-Liouville operator, $p\in C^1[a,b]$, and $u,v\in C^2[a,b]$.
I must use integration by parts (followed by differentiation), but I have no idea what the "parts" are I need to integrate by parts.
Since this is a homework problem, please do not solve the whole problem. Please just help me get the integral set up. Thanks!
I'm assuming $$ Sf = -\frac{1}{w}\frac{d}{dx}\left(p(x)\frac{df}{dx}\right). $$ The inner product is a weighted inner product with weight $w$. That suggests considering $$ \int (uSv) w dx = -\int u(pv')'dx \\ = -u(pv')+\int u'(pv')dx \\ = -u(pv')+\int(pu')v'dx \\ = -u(pv')+(pu')v-\int(pu')'vdx \\ = -u(pv')+(pu')v+\int (Su)vw dx \\ = p(u'v-uv')+\int (Su)vwdx $$