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

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

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

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

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.

External link

Riccati Equation at EqWorld: The World of Mathematical Equations.

Bibliography

A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003.

Riccati equation

External link

Bibliography

See also: Riccati equation, Bernoulli differential equation, Jacopo Francesco Riccati, Mathematics, Ordinary differential equation