Riccati equation

It is a nonlinear first order ODE that was developed in the context of hydrodynamics. It is of the form

y^{'} = P (x) + Q (x) y + R (x) y^{2},

where $P (x), Q (x), R (x)$ are known (sufficiently differentiable) functions. The typical solution method follows these steps:

Find a particular solution $y_{1}$ , often by trial and error.
Substitute $y = y_{1} + u$ into the Riccati equation, which transforms it into a Bernoulli equation in terms of $u$ . Solve this equation to obtain a one-parameter family of solutions.
The general solution is then given by $y = y_{1} + u$ .

In the particular case of the Riccati equation

y^{'} - \frac{α}{a + b} (1 - y) (a + b y) = 0.

where $a, b,$ and $α$ are real parameters the function

y = \frac{1 - a e^{- α x}}{1 + b e^{- α x}},

is a solution of the equation.

There is a relation with Schrodinger equation, I think that Riccati equations are what satisfies the exponent of a solution of a Schrodinger equation...

A Riccati-type equation can be transformed into a second order linear homogeneous differential equation (see this). Moreover, it can be "approximately transformed" into a Schrodinger equation (see this).