Riccati equation

A Riccati equation is any differential equation that has the form

It is named after Count Jacopo Francesco Riccati (1676-1754).

The Riccati equation is not amenable to elementary techniques in solving differential equations, except as follows. If one can find any solution y₁, the general solution is obtained as

y = y₁ + u

Substituting

y₁ + u

in the Riccati equation yields

and since

or

which is a Bernoulli equation. Unfortunately, one finds y₁ by guessing. The substitution that is needed to solve this Bernoulli equation is

Substituting

directly into the Riccati equation yields the linear equation

The general solution to the Riccati equation is then given by

where z is the general solution to the aforementioned linear equation.