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.