Weighted energy estimate for two coupled modes ($N=c|A{|}^2+a|B{|}^2$)

Suppose two complex amplitudes satisfy ${\partial }_t|A{|}^2\le 2a|A||B|-2d|A{|}^2$ and ${\partial }_t|B{|}^2\le 2c|A||B|-2d|B{|}^2$, with positive $a,c$ and $d\ge 0$. Then $N=c|A{|}^2+a|B{|}^2\ge 2\sqrt{ac}|A||B|$ gives $\dot{N}\le 2(\sqrt{ac}-d)N$. A positive weighted norm avoids treating two coupling rates of different size as a single arithmetic sum. This is useful for nonautonomous linear ordinary differential equations and the bounded-modulation alpha-Omega growth estimate.