Perturbative Methods for Modelling (M2)

Example 5, p. 490

We consider the Sturm-Liouville problem $\fs2 y''+E\,(x+\pi)^4\,y=0$ with the boundary conditions $\fs2 y(0)=y(\pi)=0$. Using the WKB theory, show that $\fs2 E_n\sim {9\,n^2\over 49\,\pi^4}$ and  $\fs2 y_n(x) \sim \sqrt{6\over7\,\pi^3} \,{\sin\left[n\,\left(x^3+3\,x^2\,\pi+3\,\pi^2\,x\right)/(7\,\pi^2)\right]\over(\pi+x)}$ for $\fs2 n\to\infty$.