Liouville transformation of a second-order equation
= Liouville transformation of a second-order equation
{c}
{title2=$U=\dot x^{-1/2}u(x(t))$}
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.