= Solution
Away from the turning points, the <Liouville–Green approximation> gives in the allowed region
$$
\psi\sim q^{-1/4}\left[A\cos\left(\lambda\int_a^x\sqrt q\,ds\right)
+B\sin\left(\lambda\int_a^x\sqrt q\,ds\right)\right].
$$
The bound-state condition discards the growing exponential in each forbidden region, leaving
$$
\psi_L\sim C_L(-q)^{-1/4}\exp\left[-\lambda\int_x^a\sqrt{-q}\,ds\right],\qquad
\psi_R\sim C_R(-q)^{-1/4}\exp\left[-\lambda\int_b^x\sqrt{-q}\,ds\right].
$$
These approximations require variation slow compared with the local wavelength, and must be replaced near $a,b$ by Airy approximations.
Near $a$, set $z=-\lambda^{2/3}q'(a)^{1/3}(x-a)$. The local differential equation becomes $\psi_{zz}-z\psi=0$. Decay on the left selects <Airy function> $\operatorname{Ai}(z)$ rather than its growing companion. Comparing the supplied asymptotics across the turning point yields
$$
\psi\sim2C_Lq^{-1/4}\cos\left(S_a(x)-\frac\pi4\right),
\qquad S_a(x)=\lambda\int_a^x\sqrt q\,ds.
$$
At $b$, use $z=\lambda^{2/3}[-q'(b)]^{1/3}(x-b)$; decay on the right similarly yields $2C_Rq^{-1/4}\cos(S_b(x)-\pi/4)$, where $S_b=\lambda\int_x^b\sqrt q\,ds$. Let $J=S_a+S_b=\lambda\int_a^b\sqrt q\,ds$. The left expression has equal cosine and sine coefficients in the variable $S_a$; the right expression has those coefficients proportional to $\cos(J-\pi/4)$ and $\sin(J-\pi/4)$. They can describe the same nonzero wave only when $J-\pi/4=\pi/4+n\pi$. Therefore the <WKB quantization condition> is
$$
\boxed{\lambda\int_a^b\sqrt{q(x)}\,dx=(n+\tfrac12)\pi,
\qquad n=0,1,2,\ldots.}
$$
The two simple turning points contribute the two quarter-phase shifts, giving the half-integer correction.
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