Riccati equation

From Bvio.com

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 Bernoulli equation. Unfortunately, one finds <math>y_1</math> by guessing. The substitution that is needed to solve this Bernoulli equation is

<math> z = u^{1-2} = \frac{1}{u} </math>

Substituting

<math> y = y_1 + \frac{1}{z} </math>

directly into the Riccati equation yields the linear equation

<math> z' + (q_1 + 2 \, q_2 \, y_1) \, z = -q_2 </math>

The general solution to the Riccati equation is then given by

<math> y = y_1 + \frac{1}{z} </math>

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