The differential equation dy/dx = P(x) + Q(x)y + R(x)y² is known as Riccati’s equation.

(a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution y₁ of the equation. Show that the substitution y = y₁ + u reduces Riccati’s equation to a Bernoulli equation (4) with n = 2. The Bernoulli equation can then be reduced to a linear equation by the substitution w = u^-1.

(b) Find a one-parameter family of solutions for the differential equation

dy/dx = -4/x² – 1/x y + y²

where y₁ = 2yx is a known solution of the equation.