# Riccati equation

In mathematics, a Riccati equation is any ordinary differential equation that has the form

$y' = q_0(x) + q_1(x) \, y + q_2(x) \, y^2$

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 $y_1$, the general solution is obtained as

$y = y_1 + u$

Substituting

$y_1 + u$

in the Riccati equation yields

$y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,$

and since

$y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2$
$u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2$

or

$u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2,$

which is a Bernoulli equation. Unfortunately, one finds $y_1$ by guessing. The substitution that is needed to solve this Bernoulli equation is

$z = u^{1-2} = \frac{1}{u}$

Substituting

$y = y_1 + \frac{1}{z}$

directly into the Riccati equation yields the linear equation

$z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2$

The general solution to the Riccati equation is then given by

$y = y_1 + \frac{1}{z}$

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