 Ermakov-Pinney equation

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.