Riccati Equation

A Riccati equation is a first-order nonlinear ODE of the form

$$\frac{dy}{dx}=P(x)Y62+Q(x)y+R(x)$$

where $P(x)\ne 0$

Conversion to Bernoulli Equation

If one particular solution $\varphi (x)$ is known, then the substitution

$$y=\varphi +z(x)$$

reduces the Riccati equation to a Bernoulli equation in $z$ with exponent

$$\alpha =2$$

Consequence

Since Bernoulli equations reduce to linear ODEs, knowing a single solution $\varphi$ allows construction of the general solution of the Riccati equation.