Eigen values of the ODE $(1+x^2)y’’+2xy’+\lambda x^2y=0$.

Question

How to find eigen values of the Sturm Liouville Boundary value problem $$(1+x^2)y’’+2xy’+\lambda x^2y=0, y’(1)=0, y’(10)=0?$$ I know how to solve Sturm Liouville problem of the form $y’’+\lambda =0$ by making three cases as $\lambda=0,\lambda>0$ and $\lambda <0$. In this Problem I only know that $\lambda=0$ is an eigen value because for $\lambda=0$ there are non constant solutions. I don’t know to discuss all other eigenvalues . Please help . Thank you.

Answer 1

Define a differential operator $L$ by $$ Lf = -\frac{1}{x^2}((1+x^2)f')'. $$ Your Sturm-Liouville equation is $$ Lf = \lambda f $$ subject to the endpoint conditions $$ f'(1)=0,\;\; f'(10)=0. $$ You can solve the differential equation for $\lambda =0$: $$ Lf=0 \implies (1+x^2)f'=C \\ \implies f=C\tan^{-1}(x)+D. $$ If we assume that $f'(1)=0$, then $C=0$, and the solution is $f(x)=D$. This function satisfies $Lf=0$ subject to $f'(1)=0=f'(10)$. So $f$ is an eigenfunction with eigenvalue $0$.

Next, solve a sequence of problems: $$ Lf_0 = 0,\;\; f_0(1)=1,\;f_0'(1)=0, \\ Lf_1 = f_0,\;\; f_1(1)=0,\;f_1'(1)=0, \\ Lf_2 = f_1,\;\; f_2(1)=0,\; f_2'(1)=0, \\ Lf_3 = f_2,\;\; f_3(1)=0,\; f_3'(0)=0, \\ \cdots\cdots $$ Then $$ f_{\lambda}(x)= \sum_{n=0}^{\infty}\lambda^n f_n(x) $$ is a solution of $$ Lf_{\lambda}=\lambda f_{\lambda},\;\; f_{\lambda}(0)=1, f_{\lambda}'(0)=0. $$ The eigenvalues of $L$ are the zeroes of the entire function $\lambda\mapsto f_{\lambda}'(10)$.