= 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 condition
$$
\boxed{\lambda_n\sim\frac{n\pi}{\int_0^1\sqrt{g(s)}ds},\qquad
u_n(x)\sim Cg(x)^{-1/4}\sin\left(\lambda_n\int_0^x\sqrt{g(s)}ds\right)}.
$$
Here $n$ 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.
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