The differential equation 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
where is a known solution of the equation.
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