= Solution
Let $X=\epsilon x$ and suppose $k^2(X)$ stays positive and sufficiently smooth in the region considered. Write $y=\exp[\epsilon^{-1}S_0(X)+S_1(X)+\cdots]$ in $\epsilon^2y_{XX}+k^2(X)y=0$. The leading and first transport equations of the <WKB approximation> are
$$
(S_0')^2+k^2=0,\qquad 2S_0'S_1'+S_0''=0.
$$
With $K=\sqrt{k^2}>0$, they give $S_0'=\pm iK$ and $S_1=-\tfrac12\log K+\text{constant}$. Thus the leading <WKB approximation for a slowly varying oscillator> is
$$
\boxed{y(x)\sim K(\epsilon x)^{-1/2}\left[C_+e^{i\int^xK(\epsilon s)\,ds}+C_-e^{-i\int^xK(\epsilon s)\,ds}\right].}
$$
The amplitude is fixed by the transport equation, not just by replacing the constant <wave number> in a sinusoid. The local validity condition is that variation of $K$ over a wavelength is small, for example $|dK/dx|/K^2\ll1$.
If $k^2=-\kappa^2<0$, the same calculation gives the evanescent form
$$
\boxed{y(x)\sim\kappa(\epsilon x)^{-1/2}\left[C_+e^{\int^x\kappa(\epsilon s)\,ds}+C_-e^{-\int^x\kappa(\epsilon s)\,ds}\right].}
$$
At a <classical turning point> where $k^2$ vanishes, these amplitudes diverge and the separated WKB descriptions cease to be uniform. For a simple zero, an <Airy turning-point connection formula> replaces them in an $O(\epsilon^{2/3})$ interval in $X$; it connects oscillation to exponential behavior with the usual quarter-phase shift. A higher-order zero needs a different local canonical problem, so the Airy statement assumes a simple turning point.
For a smooth positive weight $r$ on the closed interval, let $T=\int_0^\pi\sqrt{r(x)}\,dx$. The local <wave number> in the <Sturm-Liouville problem> is $\sqrt\lambda\sqrt r$, and the condition at zero selects a sine. Imposing the condition at $\pi$ gives the <WKB quantization condition>
$$
\sqrt{\lambda_n}\,T\sim n\pi,\qquad\boxed{\lambda_n\sim\left(\frac{n\pi}{T}\right)^2,\quad y_n(x)\sim C_n r(x)^{-1/4}\sin\left[\frac{n\pi}{T}\int_0^x\sqrt{r(s)}\,ds\right],\quad n\to\infty.}
$$
For weighted normalization $\int_0^\pi r|y_n|^2dx=1$, rapidly oscillating averaging gives $C_n\sim\sqrt{2/T}$. There is no turning-point phase correction here because $r$ stays positive at both endpoints with <Dirichlet boundary conditions>.
In the specified example $r(x)=(x+1)^{-2}$, so $T=\log(1+\pi)=L$ and $\lambda_n^{\rm WKB}=(n\pi/L)^2$. To solve exactly, set $t=\log(x+1)$ and $y=\sqrt{x+1}\,w(t)$. Direct differentiation reduces the <Euler-Cauchy equation> to $w''+(\lambda-1/4)w=0$ on $0<t<L$, with zero endpoint values. Hence the <inverse-square weighted Dirichlet spectrum> is
$$
\boxed{\lambda_n^{\rm exact}=\frac14+\left(\frac{n\pi}{L}\right)^2,\qquad y_n(x)=C_n\sqrt{x+1}\sin\left[\frac{n\pi\log(x+1)}L\right],\quad n=1,2,\ldots.}
$$
Values $\lambda\leq1/4$ give no nonzero solution satisfying both endpoint conditions. The WKB <eigenfunction> shape after leading quantization is actually identical to this exact shape; the leading <eigenvalue> omits $1/4$.
Using the exact <eigenvalue> as the reference, its relative error is
$$
\frac{\lambda_n^{\rm exact}-\lambda_n^{\rm WKB}}{\lambda_n^{\rm exact}}=\frac1{1+4(n\pi/L)^2}<0.01\quad\Longleftrightarrow\quad n>\frac{\sqrt{99}\,L}{2\pi}\simeq2.25.
$$
Thus \b[$n\geq3$ is sufficient and necessary among positive integers]. Referencing the approximate value instead gives $n>5L/\pi$, and the same integer answer.
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