Liouville transformation of a second-order equation ($U={\dot{x}}^{-1/2}u(x(t))$)

A smooth invertible change of variables in $u_{xx}-f(x)u=0$ introduces a first derivative in time. Rescaling by ${\dot{x}}^{-1/2}$ removes it. Choosing $\dot{x}=f^{-1/2}$ locally for $f>0$ produces the constant leading coefficient used in the WKB approximation; negative $f$ requires a complex branch or the corresponding oscillatory real formulation.