 Solution

Under the usual smoothness assumptions for WKB approximation, the strict positive lower bound on $g$ excludes turning point. Put $S(x)={\int }_0^x\sqrt{g(s)}ds$. The oscillatory approximation is $g(x{)}^{-1/4}[A\sin (\lambda S(x))+B\cos (\lambda S(x))]$. The zero boundary value at the origin forces $B=0$, and the one at the right end gives the quantization conditionHere $n$ is large and $C$ is an arbitrary normalization constant. Nonzero solutions exist only at the eigenvalues, not for every arbitrary large $\lambda$. The approximation concerns the leading phase and envelope.