Consider the regular Sturm-Liouville eigenvalue problem \left\{\begin{array}{l} -\left(x u_{x}\right)_{x}+\frac{2 u}{x}=\lambda \frac{u}{x} \text { for } 1<x<e^{x}, \\ u^{\prime}(1)=u^{\prime}\left(e^{\pi}\right)=0 . \end{array}\right. ) Determine the eigenvalue problem solved by v(y), where we define v(y)=u(x) \quad \text { with } y=\ln x \text {. } \phi_{n}(x)=\frac{\sqrt{2}}{\sqrt{\pi}} \cos (n \ln x) .

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7