 Solution

Set $z=\varphi -it$ with real constant $\varphi$. ThenThus $B=\tan (\varphi -it)/(2\hslash )$ solves the Riccati equation wherever it is finite. Integrating the amplitude equation also giveswith the square root chosen continuously from $t=0$. For the normalizable solutions used in part (c), $\sin (2\varphi )>0$; equivalently one can choose $0<\varphi <\pi /2$ modulo $\pi$. The unrestricted real parameter solves the differential equation locally, but not every value defines a normalizable wavefunction.