Research Article
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Year 2022, , 757 - 774, 30.12.2022
https://doi.org/10.18185/erzifbed.1009661

Abstract

Supporting Institution

YOK

Project Number

YOK

Thanks

YOK

References

  • [1] Berkovic, L.M. 1966. ‘‘The Reduction of Linear Ordinary Differential Equations to Equations with Constant Coefficients’’, Volz.Mat. Sb.5, 38-44.
  • [2] Boffa, V., Bollanti, S., Dattoli, G and Torre, A. 1983. ‘‘Second-Order Differential Equations with Variable Coefficients: Analytical Solutions’’, IL NUOVO CIMENTO, 99(1), 53-60.
  • [3] Breuer, S and Gottlieb, D. 1970. ‘‘The Reduction of Linear Ordinary Differential Equations to Equations with Constant Coefficients’’, Journal of Mathematical Analysis and Applications, 32, 62-76.
  • [4] Al-Hwawcha, L. K. and Abid, N.A. 2008. ‘‘A New Approach for Solving Second Order Ordinary Differential Equations’’, Journal of Mathematics and Statistics,4(1), 58-59.
  • [5] Munasinghe, R.2004. ‘‘Some Linear Differential Equations Forget That They Have Variable Coefficients’’, The College Mathematics Journal, 35(1), 22-25, DOI:10.1080/07468342.2004.11922046.
  • [6] Saravi, M. 2012. ‘‘A Procedure for Solving Some Second Order Linear Ordinary Differential Equations’’, Applied Mathematics Letters, 25, 408-411.
  • [7] Takayama, Ken.1986. ‘‘A Class of Solvable Second-Order Ordinary Differential Equations with Variable Coefficients’’, Journal of Mathematical Physics, 27(7), DOI: 10.1063-1.527038, 1747-1749.
  • [8] Urdaletova, A.B. and Kydyraliev, S.K. 1996. ‘‘Solving Linear Differential Equations by Operator Factorization’’, The College Mathematics Journal, 27(3), 199-203.
  • [9] Robin, W. 2007. ‘‘Operator factorization and the Solution of Second Order Linear Ordinary Differential Equations’’, Int. Journal of Mathematical Education in Science and Technology, 38(2), 189-211, DOI: 10.1080/00207390601002815.
  • [10] Wilmer III, A. and Costa, G.B.2008. ‘‘Solving Second Order Differential Equations with Variable Coefficients’’, Int. Journal of Mathematical Education in Science and Technology,39(2), 238-243, DOI: 10.1080/00207390701464709.
  • [11] Zraiqat, A. and Al-Hwawcha, L. K. 2015. ‘‘On Exact Solutions of Second Order Nonlinear Ordinary Differential Equations’’, Applied mathematics, 6, 953-957.
  • [12] Allen, J.L and Stein, F.M. 1964. ‘‘On Solutions of Certain Riccati Differential Equations’’, The American Math. Monthly, U.S.A., 1113-1115.
  • [13] Pala, Y. (2013). ‘‘Modern Uygulamalı Diferensiyel Denklemler (In Turkish)’’, Nobel Publications, Ankara, ISBN: 978-605-133-654-1, 63-69.
  • [14] Rao,P.R.P and Ukidave, V.H.1968. ‘‘Some Separable forms of the Riccati Equation’’, The American Mathematical Monthly, Vol.75, U.S.A.,38-39.
  • [15] Siller, H. 1970. ‘‘On the Separability of the Riccati Differential Equation’’, Mathematics Magazine, 43(4), U.S.A., 197-202.
  • [16] Pala, Y and Ertas, M.O. 2017. "A New Analytical Method for Solving General Riccati Equation," Universal Journal of Applied Mathematics, 5(2), 11-16, DOI: 10.13189/ujam.2017.050201.

New Analytical Method For Solution of Second Order Ordinary Differential Equations With Variable Coefficients

Year 2022, , 757 - 774, 30.12.2022
https://doi.org/10.18185/erzifbed.1009661

Abstract

In this study, we propose four methods for the analytical solution of second order ordinary non-homogeneous differential equations with variable coefficients. In the first case, the method directly gives the solution in explicit or integral form. In the second method, the solution of the problem reduces to the solutions of adjoined second order ordinary differential equation of homogeneous type. As long as the analytical solution of two adjoined equations can be solved, the analytical solution can always be found. In the third method, the differential equation is transformed into Riccati equation. Riccati equation is solved by means of a method recently developed. In order to solve non-homogeneous differential equation, the fourth method is developed. The strategy is different but the solution is again based on the solution of Riccati equation.

Project Number

YOK

References

  • [1] Berkovic, L.M. 1966. ‘‘The Reduction of Linear Ordinary Differential Equations to Equations with Constant Coefficients’’, Volz.Mat. Sb.5, 38-44.
  • [2] Boffa, V., Bollanti, S., Dattoli, G and Torre, A. 1983. ‘‘Second-Order Differential Equations with Variable Coefficients: Analytical Solutions’’, IL NUOVO CIMENTO, 99(1), 53-60.
  • [3] Breuer, S and Gottlieb, D. 1970. ‘‘The Reduction of Linear Ordinary Differential Equations to Equations with Constant Coefficients’’, Journal of Mathematical Analysis and Applications, 32, 62-76.
  • [4] Al-Hwawcha, L. K. and Abid, N.A. 2008. ‘‘A New Approach for Solving Second Order Ordinary Differential Equations’’, Journal of Mathematics and Statistics,4(1), 58-59.
  • [5] Munasinghe, R.2004. ‘‘Some Linear Differential Equations Forget That They Have Variable Coefficients’’, The College Mathematics Journal, 35(1), 22-25, DOI:10.1080/07468342.2004.11922046.
  • [6] Saravi, M. 2012. ‘‘A Procedure for Solving Some Second Order Linear Ordinary Differential Equations’’, Applied Mathematics Letters, 25, 408-411.
  • [7] Takayama, Ken.1986. ‘‘A Class of Solvable Second-Order Ordinary Differential Equations with Variable Coefficients’’, Journal of Mathematical Physics, 27(7), DOI: 10.1063-1.527038, 1747-1749.
  • [8] Urdaletova, A.B. and Kydyraliev, S.K. 1996. ‘‘Solving Linear Differential Equations by Operator Factorization’’, The College Mathematics Journal, 27(3), 199-203.
  • [9] Robin, W. 2007. ‘‘Operator factorization and the Solution of Second Order Linear Ordinary Differential Equations’’, Int. Journal of Mathematical Education in Science and Technology, 38(2), 189-211, DOI: 10.1080/00207390601002815.
  • [10] Wilmer III, A. and Costa, G.B.2008. ‘‘Solving Second Order Differential Equations with Variable Coefficients’’, Int. Journal of Mathematical Education in Science and Technology,39(2), 238-243, DOI: 10.1080/00207390701464709.
  • [11] Zraiqat, A. and Al-Hwawcha, L. K. 2015. ‘‘On Exact Solutions of Second Order Nonlinear Ordinary Differential Equations’’, Applied mathematics, 6, 953-957.
  • [12] Allen, J.L and Stein, F.M. 1964. ‘‘On Solutions of Certain Riccati Differential Equations’’, The American Math. Monthly, U.S.A., 1113-1115.
  • [13] Pala, Y. (2013). ‘‘Modern Uygulamalı Diferensiyel Denklemler (In Turkish)’’, Nobel Publications, Ankara, ISBN: 978-605-133-654-1, 63-69.
  • [14] Rao,P.R.P and Ukidave, V.H.1968. ‘‘Some Separable forms of the Riccati Equation’’, The American Mathematical Monthly, Vol.75, U.S.A.,38-39.
  • [15] Siller, H. 1970. ‘‘On the Separability of the Riccati Differential Equation’’, Mathematics Magazine, 43(4), U.S.A., 197-202.
  • [16] Pala, Y and Ertas, M.O. 2017. "A New Analytical Method for Solving General Riccati Equation," Universal Journal of Applied Mathematics, 5(2), 11-16, DOI: 10.13189/ujam.2017.050201.
There are 16 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Makaleler
Authors

Yaşar Pala 0000-0003-0358-1958

Çağlar Kahya 0000-0002-0722-7094

Project Number YOK
Publication Date December 30, 2022
Published in Issue Year 2022

Cite

APA Pala, Y., & Kahya, Ç. (2022). New Analytical Method For Solution of Second Order Ordinary Differential Equations With Variable Coefficients. Erzincan University Journal of Science and Technology, 15(3), 757-774. https://doi.org/10.18185/erzifbed.1009661