Ermakov-Pinney equation ($A^{\prime \prime }+p(x)A=c/A^3$)

This nonlinear ordinary differential equation governs the amplitude of a complex oscillator solution. Independent real solutions of $y^{\prime \prime }+p(x)y=0$ can be combined into a nonvanishing complex solution; its modulus satisfies the equation with $c$ equal to the squared Wronskian. A real positive amplitude and a conserved phase flux give an exact representation, not merely a WKB approximation.

Substitution of $y=A{e}^{i{ϵ}^{-1}\int \sigma dX}$ into ${ϵ}^2{y}^{\prime \prime }+k^{2}y=0$ gives the displayed real amplitude equation and conserved imaginary-part flux. Conversely these equations produce an exact complex solution wherever $A\ne 0$. The WKB approximation is obtained by expanding this exact system where $|k|$ stays positive.