Ermakov-Pinney equation
= Ermakov-Pinney equation
{c}
{title2=$A''+p(x)A=c/A^3$}
This nonlinear <ordinary differential equation> governs the amplitude of a complex oscillator solution. Independent real solutions of $y''+p(x)y=0$ 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>.