Eigen values of the ODE $(1+x^2)y’’+2xy’+\lambda x^2y=0$. 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.
 A: 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 $\lambda\mapsto f_{\lambda}'(10)$.
