Let $L$ be the operator :
$$Ly=-(py')' +ry \quad \text{with} \quad p,r>0 \quad \text{on} \quad [a,b]$$
And boundary conditions $$B_a(y)= \alpha y(a)- \beta y'(a)= 0; \quad B_b(y)= \theta y(b) + \delta y'(b)=0 \quad \alpha,\delta,\theta,\beta>0 $$
$1)$Prove that $L$ is a positive symmetric operator(do the calculations).
$2)$Prove that eigenspaces are onedimensional. Hint: write the equation as a system and calculate the Wronskian determinant in an appropriate point.
My attempt:
So to show that the operator is symmetric we need to show that $$\langle Ly,z\rangle = \langle Az,y\rangle$$
Hence let's start $$\langle Ly,z\rangle = \int( -(py')' +ry)z \, dx= \int -(py')'z + ryz\, dx $$
On the other hand $$\langle Lz,y\rangle = \int( -(pz')' +rz)y \, dx= \int -(pz')'y + ryz\, dx $$
$$ \int ryz\, dx = \int rzy\, dx$$, so we must show
$$ \int -(pz')'y = \int -(py')'z$$
Can anyone help how to proceed from now?