Riccati Equation and ODEs: An Example

Universal journal of mathematics and applications, 2023

We give a useful and practicable solution method for the general Riccati differential equation of the form $w^{\prime }\left( x\right) =p\left( x\right) +q\left( x\right) w\left( x\right) +r\left( x\right) w^{2}\left( x\right) $. In order to get the general solution many authors have been interested this type equation. They show that if there exists some relation about the coefficients $p\left( x\right),$ $q\left( x\right),$ and $r\left( x\right) $ then the general solution of this equation can be given in a closed form. We also determine some relations between these coefficients and find the general solutions to the given equation. Finally, we give some examples to illustrate the importance of the presented method.

AIP Conference Proceedings

In this article, we provide some formulas for analytical general as well as a particular solution of two classes of Riccati equation using the analytical general solution of a specific class of linear ordinary differential equations (ODEs) of second-order obtained by an alternative analytical approach. We will also solve some Riccati equations.

Journal of Physics A: Mathematical and Theoretical, 2007

We present several second-order linear differential equations that are associated to a particular Riccati equation with only one constant parameter in its coefficients through the technique of supersymmetric factorizations and through a Dirac-like procedure. The latter approach is a minimal extension of the results obtained with the first technique in the sense that it includes up to two more constant parameters.

Journal of Taibah University for Science

In this article, an operational matrix approach is presented to solve the Riccati type differential equations with functional arguments. These equations are encountered in Mathematical Physics. The method is based on the least-squares approximation and the operational matrices of integration and product. By obtaining the operation matrices for each term of the problem, the method converts the problem to a system of nonlinear algebraic equations. The roots of last system are used in determination of unknown function. Error analysis is made. Numerical applications are given to show efficiency of the method and also the comparisons are made with other methods from literature. In applications of the method, it is observed from the applications that the suggested method gives effective results.

Universal Journal of Applied Mathematics

In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Ten new exact solutions of the Riccati equation dy/dx = a(x) + b(x)y + c(x)y 2 are presented. The solutions are obtained by assuming certain relations among the coefficients a(x), b(x) and c(x) of the Riccati equation, in the form of some integral or differential expressions, also involving some arbitrary functions. By appropriately choosing the form of the coefficients of the Riccati equation, with the help of the conditions imposed on the coefficients, we obtain ten new integrability cases for the Riccati equation. For each case the general solution of the Riccati equation is also presented. The possibility of the application of the obtained mathematical results for the study of anisotropic general relativistic stellar models is also briefly considered.

In this paper, a numerical technique-differential transform method (DTM) is presented for the solution of Riccati differential equations. Three illustrative examples involving both constant and variable coefficients are solved. The DTM applied provides results that converge rapidly to the exact solution. To see the accuracy of this method, the results are compared with the exact solutions.

iaset, 2020

Riccati differential equation is one of the most essential tools for modelling many physical situations, such as spring mass systems, resistor-capacitor-induction circuits, and chemical reactions among many others. It is applicable in engineering and science, and also useful in network synthesis and optimal control. We derived a quarter-step method for the solution of RDEs by collocating and interpolating the Laguerre polynomial basis function which does not require starting values before they are implemented and they simultaneously generate approximations at different grid points in the interval of integration. To show the accuracy and efficiency of our method, five (5) model RDE problems were solved and results obtained in terms of the point wise absolute errors shows that the method approximates well with the exact solution. The stability analysis conducted reveals that our method is zero-stable, consistent and convergent.

Journal of Physics A: Mathematical and Theoretical, 2008

It has been proven by that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions are easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a 'growth' parameter from the trivial null solution up to the particular solution found through the factorization procedure.