Sturm-Liouvile Boundary Value Problems

Definition

$[p(x)y^{\prime }{]}^{\prime }-q(x)y+\lambda r(x)y=0,0<x<1$ $a_{1}y(0)-a_2{y}^{\prime }(0)=0,b_{1}y(1)-b_2{y}^{\prime }(1)=0$

Facts

The differential operator $L[y]=-[p(x)y^{\prime }{]}^{\prime }+q(x)y\Rightarrow L[y]=\lambda r(x)y$
We assume that the functions $p,p^{\prime },q,r$ are continuous on $0\le x\le 1$
and $p(x)>0,r(x)>0$ on $[0,1]$

Lagrange identity: ${\int }_0^{1}L[u]vdx={\int }_0^{1}L[v]udx\iff (L[u],v)=(u,L[v])$

All of the eigenvalues and corresponding eigen functions are real