AM  Vol.11 No.7 , July 2020
Laplace Transform, Non-Constant Coefficients Differential Equations and Applications to Riccati Equation
Abstract: In this paper, the Laplace Transform is used to find explicit solutions of a fam-ily of second order Differential Equations with non-constant coefficients. For some of these equations, it is possible to find the solutions using standard tech-niques of solving Ordinary Differential Equations. For others, it seems to be very difficult indeed impossible to find explicit solutions using traditional methods. The Laplace transform could be an alternative way. An application on solving a Riccati Equation is given. Recall that the Riccati Equation is a non-linear differential equation that arises in many topics of Quantum Me-chanics and Physics.
Cite this paper: Ndiaye, M. (2020) Laplace Transform, Non-Constant Coefficients Differential Equations and Applications to Riccati Equation. Applied Mathematics, 11, 639-649. doi: 10.4236/am.2020.117043.

[1]   Dyke, P. (2014) An Introduction to Laplace Transform and Fourier Series. Springer-Verlag, London.

[2]   Schiff, J.L. (1999) The Laplace Transform Theory and Applications. Springer-Verlag, New York.

[3]   Stroud, K.A. (2003) Advanced Engineering Mathematics.

[4]   Dass, H.K. (2009) Advanced Engineering Mathematics. S.Chand and Company Ltd., New Delhi.

[5]   Blanchard, P., Devaney, R.L. and Hall, G.R. (2012) Differential Equations. 4th Edition. Brooks/Cole, Boston.

[6]   Misra, M. Laplace Transform. Hardwari Publications, Alla Habad (India).

[7]   Hassan Eltayeb, A.K. and Kilicman, A. (2010) A Note on the Sunudu Transform and Differential Equations. Applied Mathematical Sciences , 4, 167-173.

[8]   Elkazi, T.M. and Elkazi, S.M. (2011) On the Elkazi Transform and Ordinary Differential Equations with Variable Coefficients. Advanced in Theoretical and Applied Mathematics, 6, 13-18.

[9]   Elkazi, T.M., Elkazi, S.M. and Hilal, E.M.A. (2012) Elkazi and Sumudu Transforms for Solving Some Differential Equations. Global Journal of Pure and Applied Mathematics, 2, 167-173.

[10]   O’connor, J.J. and Robertson, E.F. (1996) Jacopo Francesco Riccati.

[11]   Schuch, D. (2014) Nonlinear Riccati Equations as a Unifying Link between Linear Quantum Mechanics and Other Fields of Physics. Journal of Physics: Conference Series, 504, Article ID: 012005.

[12]   Haaheim, D.R. and Stein, F.M. (1969) Methods of Solution of the Riccati Differential Equation. Mathematics Magazine, 42, 233-240.