= Solution
Work on an interval where the <change of variables> is smooth and invertible, with $s=\dot x\ne0$. If $v(t)=u(x(t))$, the <chain rule> gives $\dot v=s u_x$ and $\ddot v=s^2u_{xx}+\dot s u_x$. Eliminating $u_{xx}$ using the original <second-order linear differential equation> gives
$$
\boxed{\ddot v-\frac{\dot s}{s}\dot v-s^2f(x)v=0.}
$$
For the real transformation as printed, take $s>0$. Set $v=s^{1/2}U$, which is the <Liouville transformation of a second-order equation>. Differentiating twice makes the coefficient of $\dot U$ cancel. Dividing by $s^{1/2}$ leaves
$$
\boxed{\ddot U-\left[s^2f(x)+\frac{3\dot s^2}{4s^2}-\frac{\ddot s}{2s}\right]U=0.}
$$
The additional term is exactly $s^{-1/2}[(\dot s/s)d(s^{1/2})/dt-d^2(s^{1/2})/dt^2]$, as required.
On an interval with $f>0$, choose $s=f^{-1/2}$ and therefore $t=\int^x\sqrt{f(\zeta)}\,d\zeta$ up to a constant. Primes in the following expression mean derivatives with respect to $x$. We have $\dot s=-f'/(2f^2)$ and $\ddot s=-f''/(2f^{5/2})+(f')^2/f^{7/2}$, giving the <WKB correction term>
$$
\boxed{\ddot U-[1+F(t)]U=0,\qquad F(t)=\left[\frac{f''}{4f^2}-\frac{5(f')^2}{16f^3}\right]_{x=x(t)}.}
$$
Neglecting $F$ gives $U=c_+e^t+c_-e^{-t}$. Since $u=s^{1/2}U$, the <WKB approximation> is
$$
\boxed{u(x)\approx f(x)^{-1/4}\left(c_+e^{\int^x\sqrt f\,d\zeta}+c_-e^{-\int^x\sqrt f\,d\zeta}\right),\qquad A(x)=f(x)^{-1/4},\quad G(x)=\int^x\sqrt f\,d\zeta.}
$$
Changing the lower integration limit only rescales the constants. Small $|F|$ is the stated local approximation criterion; rapid variation or a turning point can invalidate it, and small local residual alone need not imply uniform accuracy over an arbitrarily long interval.
The printed hypothesis only says $f\ne0$. If real $f<0$ on the interval, the printed square roots require a consistent complex branch. Equivalently a real <WKB approximation> is
$$
u(x)\approx |f(x)|^{-1/4}\left[C_1\cos\left(\int^x\sqrt{|f|}\,d\zeta\right)+C_2\sin\left(\int^x\sqrt{|f|}\,d\zeta\right)\right].
$$
This supplies the oscillatory interpretation for the negative-$f$ case instead of silently assuming that the original nonzero hypothesis means positive. Smoothness through the derivatives used above and absence of zeros are required locally.
Back to article page
