I have a question: With what positive function must the operator $L(y)=x^2y''+y' $ be multiplied, to be self-adjoint? With what weight are then the "eigenfunctions" of the multiplied operator orthogonal, with boundary conditions y(1)=0=y(2) ?
So from the weight equation : $w(x)=\frac{1}{a(x)}e^{\int \frac{b(x)}{a(x)}dx}$
Where in general $ L y =a(x)y''+b(x)y'+c(y)$
My solution: the function should be multiplied by $\frac{1}{x^2}e^{\int 1/x^2} $ and the weight is then $\frac{1}{x^2}e^{\int 1/x^2} $ , is this correct, or am I wrong ?