Riccati's equation

y = (1/z)(dz/dx)
dy/dx = [d(1/z)/dx](dz/dx) + (1/z)(d^2z/dx^2)
dy/dx = (-1/z^2)(dz/dx)^2 + (1/z)(d^2z/dx^2)
Therefore dy/dx + y^2 + Py + Q = 0 becomes
(-1/z^2)(dz/dx)^2 + (1/z)(d^2z/dx^2) + (1/z^2)(dz/dx)^2 + (P/z)(dz/dx) + Q = 0
(1/z)(d^2z/dx^2) + (P/z)(dz/dx) + Q = 0
d^2z/dx^2 + P(dz/dx) + Qz = 0