# Riccati equation

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

[itex] y' = q_0(x) + q_1(x) \, y + q_2(x) \, y^2 [itex]

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 [itex]y_1[itex], the general solution is obtained as

[itex] y = y_1 + u [itex]

Substituting

[itex] y_1 + u [itex]

in the Riccati equation yields

[itex] y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,[itex]

and since

[itex] y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2 [itex]
[itex] u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2 [itex]

or

[itex] u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2, [itex]

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

[itex] z = u^{1-2} = \frac{1}{u} [itex]

Substituting

[itex] y = y_1 + \frac{1}{z} [itex]

directly into the Riccati equation yields the linear equation

[itex] z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2 [itex]

The general solution to the Riccati equation is then given by

[itex] y = y_1 + \frac{1}{z} [itex]

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

• Riccati Equation (http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf) at EqWorld: The World of Mathematical Equations.

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