An Analytical Method for Solving General Riccati Equation

Authors: Y. Pala, M. O. Ertas

Abstract:

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.

Keywords: Riccati Equation, ordinary differential equation, nonlinear differential equation, analytical solution, proper solution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1340124

References:

[1] Allen, J.L. and Stein, E.M, On the Solution of Certain Riccati Equations, The American Math. Montly, U.S.A., pp.1113-1115, 1964.
[2] Harko, T., Lobo, F.S.N., and Mak, M.K., Analytical Solution of the Riccati Equation with Coefficients Satisfying Integral or Differential Conditions with Arbitrary Functions, Universal Journal of Applied Mathematics, Vol.2, U.S.A., pp.109-118, 2014.
[3] Ince, E.L., Ordinary Differential Equation, ISBN: 978-0-486-60349-0, Dover Publications, New York-U.S.A., pp. 1-204, 1956.
[4] Kreyszig, E., Advanced Engineering Mathematics, ISBN: 0-471-33328-X, John Wiley&Sons. Inc, New York-U.S.A., pp.1-146, 1999.
[5] Mak, M.K. and Harko, T., New Integrability Case for the Riccati Equation, Applied Mathematics and Computation, Vol.218, Netherlands, pp.10974-10981, 2012.
[6] Mortici, C., The method of the variation of constants for Riccati Equations, General Mathematics Vol. 16, No. 1, Romania, pp.111-116, 2008).
[7] Pala, Y., Modern Differential Equations and Its Applications, ISBN: 075-591-936-8, Nobel Publications, Bursa-Turkey, pp.1-188, 2006.
[8] Rao, P.R.P., The Riccati Differential Equation, The American Mathematical Montly, U.S.A., pp.995-996, 1962.
[9] Rao, P.R.P. and Ukidave, V.H., Separable forms of the Riccati Equation, The American Mathematical Montly, Vol.75, U.S.A., pp.38-39, 1968.
[10] Siller, H., On the Separability of the Riccati Differential Equation, Mathematics Magazine, Vol.43, No.4, U.S.A., pp.197-202, 1970.
[11] Sugai, I., Riccati’s Nonlinear Differential Equation, The American Mathematical Monthly, Vol.67, No.2, U.S.A., pp.134-139, 1960.