EQUATION DIFFERENTIELLE DE RICCATI PDF

Riccati Equations Before we give the formal definition of Riccati equations, a little introduction may be helpful. Indeed, consider the first order differential equation If we approximate f x,y , while x is kept constant, we will get If we stop at y, we will get a linear equation. Riccati looked at the approximation to the second degree: he considered equations of the type These equations bear his name, Riccati equations. They are nonlinear and do not fall under the category of any of the classical equations. In order to solve a Riccati equation, one will need a particular solution. Without knowing at least one solution, there is absolutely no chance to find any solutions to such an equation.

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Riccati Equations Before we give the formal definition of Riccati equations, a little introduction may be helpful. Indeed, consider the first order differential equation If we approximate f x,y , while x is kept constant, we will get If we stop at y, we will get a linear equation.

Riccati looked at the approximation to the second degree: he considered equations of the type These equations bear his name, Riccati equations. They are nonlinear and do not fall under the category of any of the classical equations. In order to solve a Riccati equation, one will need a particular solution.

Without knowing at least one solution, there is absolutely no chance to find any solutions to such an equation. Indeed, let y1 be a particular solution of Consider the new function z defined by Then easy calculations give which is a linear equation satisfied by the new function z.

Once it is solved, we go back to y via the relation Keep in mind that it may be harder to remember the above equation satisfied by z. Instead, try to do the calculations whenever you can. We recognize a Riccati equation. First of all we need to make sure that y1 is indeed a solution. Otherwise, our calculations will be fruitless.

Set Then we have Hence, from the equation satisfied by y, we get Easy algebraic manipulations give.

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Migal Cauchy ricati with this approach have been studied in [21, 26]. Moreover, recent applications to mathematical physics can be found in [1, 3, 27, 28, 32]. Log In Sign Up. Suslov, Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields, Lett. This paper is organized in the following way: The special functions are not always Liouvillian, we can see that Airy equation has not Liouvillian solutions, while Bessel equation has Liouvillian solutions for special values of the parameter, see [13, 29].

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