Sturm-Liouville Problem: Find Eigenfunctions

To avoid the problem of discontinuities, I think that it could be simpler to consider that we search for the zero's of function $$f(u)=u \sin(u)+\pi \cos(u)$$ Since the $n^{th}$solution $u_n \sim n \pi$, we can develop $f(u)$ as a Taylor series around $n \pi$ and use series reversion to get $$u_n=n\pi-\frac{1}{n}+\frac{\frac{1}{3}-\frac{1}{\pi }}{n^3}+\frac{-\frac{1}{5}-\frac{2}{\pi ^2}+\frac{4}{3 \pi }}{n^5}+O\left(\frac{1}{n^7}\right)$$ The table below reports the value of the approximation as well as the exact solution obtained using Newton method. $$\left( \begin{array}{ccc} n & \text{approximation} & \text{solution} \\ 1 & 2.17838691503 & 2.17672257444 \\ 2 & 5.78574357602 & 5.78574088629 \\ 3 & 9.09209064323 & 9.09209060673 \\ 4 & 12.3166266163 & 12.3166266170 \\ 5 & 15.5080904222 & 15.5080904229 \\ 6 & 18.6829616076 & 18.6829616079 \\ 7 & 21.8483365278 & 21.8483365279 \\ 8 & 25.0077712358 & 25.0077712358 \\ 9 & 28.1632437482 & 28.1632437482 \\ 10 & 31.3159417771 & 31.3159417771 \end{array} \right)$$ Back to $\sqrt \lambda \pi=u$, this would give $$\lambda_n=n^2-\frac{2}{\pi }+\frac{2 \pi -3}{3 \pi ^2 n^2}-\frac{10-10 \pi +2 \pi ^2}{5 \pi ^3 n^4}+\frac{225-150 \pi +23 \pi ^2}{45 \pi ^4 n^6}+O\left(\frac{1}{n^8}\right)$$