Riccati equation

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

<math> y' = q_0(x) + q_1(x) \, y + q_2(x) \, y^2 <math>

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 <math>y_1<math>, the general solution is obtained as

<math> y = y_1 + u <math>

Substituting

<math> y_1 + u <math>

in the Riccati equation yields

<math> y_1' + u' = q_0 + q_1 \cdot (y_1 + u) + q_2 \cdot (y_1 + u)^2,<math>

and since

<math> y_1' = q_0 + q_1 \, y_1 + q_2 \, y_1^2 <math>

<math> u' = q_1 \, u + 2 \, q_2 \, y_1 \, u + q_2 \, u^2 <math>

or

<math> u' - (q_1 + 2 \, q_2 \, y_1) \, u = q_2 \, u^2, <math>

which is a