Let $L[y]:=y''''$. Let the domain of $L$ be the set of functions that have four continuous derivatives on $[0,π]$ and satisfy $y(0)=y'(0)=0$ and $y(π)=y'(π)=0$
a) Show that $L$ is self adjoint
b) Use the result of part a) to show that eigenfunctions corresponding to distinct eigenvalues of the problem $y''''+ λy=0$ ; $y(0)=y'(0)=0$ and $y(π)=y'(π)=0$ are orthogonal with $w(x)$ identically equal to $1$ on $[0,π]$.
for part a) I know you must show $(u,L[v])=(L[u],v)$ i.e integral from a to be of $u.L[v]=$ integral from a to be of $L[u].v$. In a previous problem from this section they chose $u(x)=sinx$ and $v(x)=sin2x$ (not sure why, in that problem $L[y]:= y" + 2y' +5y$) should I do that again to show that $L$ is self adjoint?
for part b) I know that two real-valued functions $f$ and $g$ are orthogonal with respect to a positive weight function $w(x)$ on the interval $[a,b]$ if the integral from $a$ to $b$ of $f(x)g(x)w(x)dx=0$ however I'm confused to as to what $f(x)$ and $g(x)$ are in this problem. Thanks!!!