Riccati equation
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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.
External link
- Riccati Equation (http://eqworld.ipmnet.ru/en/solutions/ode/ode0123.pdf) 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.pl:Równanie różniczkowe Riccatiego fr:Équation de Riccati