# Upper and Lower bound eigenvalues Sturm Liouville problem

1. Apr 7, 2009

### redarrows

I have 2 questions that need to be solve:

01. Find upper and lower bound for the k-th eigenvalue $$\lambda_{k}$$ of the problem $$((1+x^2)u')'-xu+\lambda(1+x^2)u$$ for $$0< x< 1$$ with boundary conditions $$u(0)=0$$ and $$u(1)=0$$

02. Find a lower bound for the lowest eigenvalue of the problem
$$((1+x^2)u')'+\lambda(1+x^2)u=0$$ for $$0 < x < 1$$ with boundary conditions $$u(0)=0$$ and $$u'(1)=0$$

many thanks and looking forward from anyone. please put the steps as well.