Differential topology

I thinks there is some problem related to the solution of a 2 order differential equation given by Sturm-Liouville system which is nontrivial. It is well-know that the power of Sturm-Liouville theory see  wiki, is due to it is some kind of “spectral decomposition” in the solution space. Two kind of problem is interesting, one is the eigenvalue estimate, both upper bound and lower bound, this already investigated in ESTIMATING THE EIGENVALUES OF STURM-LIOUVILLE. PROBLEMS BY APPROXIMAT…

https://en.wikipedia.org/wiki/Transversality_(mathematics) This problem may be a embarrassed one, but I even could not prove it for the 1 dimensional case. Here is the problem: >**Question 1** $latex M$ is a compact $latex n$-dimensional smooth manifold in $latex R^{n+1}$, take a point $p\notin M$. prove there is always a line $latex l_p$ pass $latex p$ and $latex l_p\cap M\neq \emptyset$, and $latex l_p$ intersect transversally with $latex M$. You can naturally generated it to: >**Qusetio…