Find the Eigenvalues of $xy''+y'+λy=0, y(1)=5, y(e)=2$

Question

Hi everybody I have to Find the Eigenvalues of $xy''+y'+λy=0,$ in $y(1)=5, y(e)=2$

I think it has to be in the Stourm-Liouville form: $d/dx(xy')+λy/x=0$

but Im not sure about this

Answer 1

Your terminology is a bit off, I think. The Sturm-Liouville problem in this context would be $x y'' + y' + \lambda y = 0$ with boundary values $y(1) = y(e) = 0$. An eigenvalue would be $\lambda$ such that this problem has a nonzero solution. For such $\lambda$, the differential equation with your boundary values $y(1)=5$, $y(e)=2$ might not have a solution; for every $\lambda$ that is not an eigenvalue, it will have a solution.

The fundamental solutions of $x y'' + y' + \lambda y = 0$ are $J_0(2\sqrt{\lambda x})$ and $Y_0(2\sqrt{\lambda x})$ if $\lambda \ne 0$, where $J_0$ and $Y_0$ are Bessel functions of the first and second kinds, or $1$ and $\ln(x)$ if $\lambda = 0$. In order for $\lambda$ to be an eigenvalue we need $$Y_0(2\sqrt{\lambda e} J_0(2\sqrt{\lambda}) = Y_0(2\sqrt{\lambda})J_0(2\sqrt{\lambda e})$$ That occurs for (approximately) $\lambda = 5.826546274,\; 23.41467095,\;52.72970212, \ldots$.