Eigen Values of $(\frac{1}{3x^2+1}y’)’+\lambda (3x^2+1)y=0, y(0)=0, y(\pi)=0.$

Question

How to find eigen values of Sturm Liouville Problem $$(\frac{1}{3x^2+1}y’)’+\lambda (3x^2+1)y=0, y(0)=0, y(\pi)=0?$$ I only know how to find eigen values of $y’’+\lambda y=0, y(0)=y(\pi)=0,$ because ODE $y’’+\lambda y=$ can be solve easily . I have no ideal how to find eigen values of $$(\frac{1}{3x^2+1}y’)’+\lambda (3x^2+1)y=0, y(0)=0, y(\pi)=0.$$ please help. Thank you.

Answer 1

Hint: Let $y(x)=z(\xi)$, where $\xi=x^3+x$. Show that $z(\xi)$ satisfies the ODE $z''(\xi)+\lambda z(\xi)=0$ with boundary conditions $z(0)=z(\pi^3+\pi)=0.$