|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
Riccati equation
|
(Result)
|
|
|
The nonlinear differential equation
 |
(1) |
is called Riccati equation. If
 , it is a question of a linear differential equation; if
 , of a Bernoulli equation. There is no general method for integrating explicitely the equation (1), but via the substitution
one can convert it to a second order homogeneous linear differential equation with non-constant coefficients.
If one can find a particular solution  , then one can easily verify that the substitution
 |
(2) |
converts (1) to
![$\displaystyle \frac{dw}{dx}+[g(x)\!+\!2h(x)y_0(x)]\,w+h(x) = 0,$](http://images.planetmath.org:8080/cache/objects/10635/l2h/img7.png "$\displaystyle \frac{dw}{dx}+[g(x)\!+\!2h(x)y_0(x)]\,w+h(x) = 0,$") |
(3) |
which is a linear differential equation of first order with respect to the function  .
Example. The Riccati equation
 |
(4) |
has the particular solution  . Solve the equation.
We substitute
 to (4), getting
For solving this first order equation we can put  ,
 , writing the equation as
 |
(5) |
where we choose the value of the expression in parentheses equal to 0:
After separation of variables and integrating, we obtain from here a solution
 , which is set to the equation (5):
Separating the variables yields
and integrating:
Thus we have
whence the general solution of the Riccati equation (4) is
It may be proved that if one knows three different solutions of Riccati equation (1), the each other solution may be expresses as a rational function of them.
|
"Riccati equation" is owned by pahio.
|
|
(view preamble | get metadata)
| Other names: |
Riccati differential equation |
This object's parent.
|
|
Cross-references: rational function, general solution, variables, separating, solution, separation of variables, expression, function, linear differential equation of first order, particular solution, coefficients, homogeneous, equation, Bernoulli equation, linear differential equation, nonlinear differential equation
There are 2 references to this entry.
This is version 8 of Riccati equation, born on 2008-05-28, modified 2008-10-17.
Object id is 10635, canonical name is RiccatiEquation.
Accessed 1030 times total.
Classification:
| AMS MSC: | 34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions) | | | 34A34 (Ordinary differential equations :: General theory :: Nonlinear equations and systems, general) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle w = w(x) = uv = e^{x^3}\left[C+\int xe^{-x^3}\,dx\right],$](http://images.planetmath.org:8080/cache/objects/10635/l2h/img21.png "$\displaystyle w = w(x) = uv = e^{x^3}\left[C+\int xe^{-x^3}\,dx\right],$")
