Consider the equation

$$\epsilon^{2} \frac{d^{2} y}{d x^{2}}=Q(x) y$$

where $\epsilon>0$ is a small parameter and $Q(x)$ is smooth. Search for solutions of the form

$$y(x)=\exp \left[\frac{1}{\epsilon}\left(S_{0}(x)+\epsilon S_{1}(x)+\epsilon^{2} S_{2}(x)+\cdots\right)\right],$$

and, by equating powers of $\epsilon$, obtain a collection of equations for the $\left\{S_{j}(x)\right\}_{j=0}^{\infty}$ which is formally equivalent to (1). By solving explicitly for $S_{0}$ and $S_{1}$ derive the Liouville- Green approximate solutions $y^{L G}(x)$ to (1).

For the case $Q(x)=-V(x)$, where $V(x) \geqslant V_{0}$ and $V_{0}$ is a positive constant, consider the eigenvalue problem

$$\frac{d^{2} y}{d x^{2}}+E V(x) y=0, \quad y(0)=y(\pi)=0$$

Show that any eigenvalue $E$ is necessarily positive. Solve the eigenvalue problem exactly when $V(x)=V_{0}$.

Obtain Liouville-Green approximate eigenfunctions $y_{n}^{L G}(x)$ for (2) with $E \gg 1$, and give the corresponding Liouville Green approximation to the eigenvalues $E_{n}^{L G}$. Compare your results to the exact eigenvalues and eigenfunctions in the case $V(x)=V_{0}$, and comment on this.