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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 , the general solution is obtained as

Substituting
in the Riccati equation yields
and since
or
which is a Bernoulli equation. Unfortunately, one finds 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.