Show that the Sturm Liouville problem given by $$u''+ \lambda xu=0$$ on the interval $[0,1]$ satisfying the boundary conditions $u(0)=u(1)+u'(1)=0$ has no negative eigenvalues.
I have tried computing the Rayleigh quotient $$ \frac{\int_{0}^{1} uu'' dx}{\int_{0}^{1} xu^2 dx}$$ but this gives me a negative value because of the boundary conditions and I can't prove what I want.
Any help please?
If $u$ is an eigenfunction, then $\lambda$ is real, $u$ may be assumed to real, and \begin{align} \lambda\int_{0}^{1}xu^2dx & = -\int_{0}^{1}u''udx \\ & = -u'u|_{0}^{1}+\int_{0}^{1}u'^2dx \\ & = -u'(1)u(1)+u'(0)u(0)+\int_{0}^{1}u'^2dx \\ & = u(1)^2 +\int_{0}^{1}u'^2 dx \end{align} Therefore $\lambda \ge 0$ must hold.