A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients

Mohamed Abdirahman Jama; Kang'ethe Giterere; Duncan Gathungu Kioi

doi:doi:10.11648/j.acm.20251402.11

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A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients

Mohamed Abdirahman Jama^*

, Kang'ethe Giterere, Duncan Gathungu Kioi

Published in Applied and Computational Mathematics (Volume 14, Issue 2)

Received: 26 January 2025 Accepted: 18 February 2025 Published: 5 March 2025

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Abstract

Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.

Published in	Applied and Computational Mathematics (Volume 14, Issue 2)
DOI	10.11648/j.acm.20251402.11
Page(s)	78-89
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Integral Transforms, Laplace Transform, New Analytic Method

References

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[4]	William E. Boyce, Richard C. DiPrima. “Elementary Differential Equations and Boundary Value Problems,” Hoboken, New Jersey: John Wiley and Sons, Inc; 2009. pp. 219-239.
[5]	Ahmed Mohammed, and Aklilu Zeleke, “Extending the Constant Coefficient Solution Technique to Variable Coefficient Ordinary Differential Equations,” PRIMUS. 2015, Vol. 25, no. 6, pp. 485-494. https://doi.org/10.1080/10511970.2015.1025160
[6]	Kelly, “Mechanical vibrations: theory and applications,” Boston, USA: Cengage Learning; 2012. pp. 1-672.
[7]	Said-Houari, B., “Differential Equations: Methods and Applications,” Cham, Switzerland: Springer; 2016. pp. 1-12. https://doi.org/10.1007/978-3-319-25735-8
[8]	Tung, L. J. and Kwan, B. W., “Circuit analysis,” New Jersey, USA: World Scientific Publishing; 2001. pp. 1- 253. https://doi.org/10.1142/4203
[9]	Singh, D. P. and Ujlayan, “An alternative approach to write the general solution of a class of second order linear differential equation,” Resonancel. 2021, Vol. 26, no. 5, pp. 705-714. https://doi.org/10.1007/s12045-021-1170-8
[10]	Dennis G. Zill, Michael R. Cullen, “Differential Equations with Boundary-value Problems,” Boston, USA: Cengage Learning; 2005.
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[13]	Wilmer III, A. and Costa, G., “Solving second order differential equations with variable coefficients,” International Journal of Mathematical Education in Science and Technology. 2008, Vol. 39, no. 2, pp. 238- 243. https://doi.org/10.1080/00207390701464709
[14]	Ahmed, Z. and Kalim, M, “A new transformation technique to find the analytical solution of general second order linear ordinary differential equation,” International Journal of Advanced and Applied Sciences. 2018, Vol. 5, no. 4, pp. 109-114. https://doi.org/10.21833/ijaas.2018.04.014
[15]	Beccar-Varela, M. P., Bhuiyan, M. A. M., Mariani, M. C., and Tweneboah, O. K., “Analytic methods for solving higher order ordinary differential equations,” Mathematics. 2019, Vol. 7, no. 9, pp. 826. https://doi.org/10.3390/math7090826
[16]	Pala, Y. and Kahya, “New analytical methodfor solution of second order ordinary differential equations with variable coefficients,” Erzincan University Journal of Science and Technology. 2022, Vol. 15, no. 3, pp. 757- 774. https://doi.org/10.18185/erzifbed.1009661
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APA Style

Jama, M. A., Giterere, K., Kioi, D. G. (2025). A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients. Applied and Computational Mathematics, 14(2), 78-89. https://doi.org/10.11648/j.acm.20251402.11

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ACS Style

Jama, M. A.; Giterere, K.; Kioi, D. G. A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients. Appl. Comput. Math. 2025, 14(2), 78-89. doi: 10.11648/j.acm.20251402.11

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AMA Style

Jama MA, Giterere K, Kioi DG. A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients. Appl Comput Math. 2025;14(2):78-89. doi: 10.11648/j.acm.20251402.11

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@article{10.11648/j.acm.20251402.11,
  author = {Mohamed Abdirahman Jama and Kang'ethe Giterere and Duncan Gathungu Kioi},
  title = {A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {2},
  pages = {78-89},
  doi = {10.11648/j.acm.20251402.11},
  url = {https://doi.org/10.11648/j.acm.20251402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251402.11},
  abstract = {Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.},
 year = {2025}
}

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TY  - JOUR
T1  - A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients
AU  - Mohamed Abdirahman Jama
AU  - Kang'ethe Giterere
AU  - Duncan Gathungu Kioi
Y1  - 2025/03/05
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251402.11
DO  - 10.11648/j.acm.20251402.11
T2  - Applied and Computational Mathematics
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AB  - Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.
VL  - 14
IS  - 2
ER  -

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Author Information

Mohamed Abdirahman Jama

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0008-8289-5763
Kang'ethe Giterere

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
Duncan Gathungu Kioi

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

http://orcid.org/0000-0002-4447-2734

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Plain Text BibTeX RIS

APA Style

Jama, M. A., Giterere, K., Kioi, D. G. (2025). A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients. Applied and Computational Mathematics, 14(2), 78-89. https://doi.org/10.11648/j.acm.20251402.11

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ACS Style

Jama, M. A.; Giterere, K.; Kioi, D. G. A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients. Appl. Comput. Math. 2025, 14(2), 78-89. doi: 10.11648/j.acm.20251402.11

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AMA Style

Jama MA, Giterere K, Kioi DG. A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients. Appl Comput Math. 2025;14(2):78-89. doi: 10.11648/j.acm.20251402.11

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@article{10.11648/j.acm.20251402.11,
  author = {Mohamed Abdirahman Jama and Kang'ethe Giterere and Duncan Gathungu Kioi},
  title = {A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {2},
  pages = {78-89},
  doi = {10.11648/j.acm.20251402.11},
  url = {https://doi.org/10.11648/j.acm.20251402.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251402.11},
  abstract = {Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - A Novel Analytic Method with Integral Transform for Solving Classes of Second and Third Order Ordinary Linear Differential Equations with Variable Coefficients
AU  - Mohamed Abdirahman Jama
AU  - Kang'ethe Giterere
AU  - Duncan Gathungu Kioi
Y1  - 2025/03/05
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251402.11
DO  - 10.11648/j.acm.20251402.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 78
EP  - 89
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251402.11
AB  - Analytical solutions of second- and third-order non-homogeneous Ordinary Linear Differential Equations (OLDEs) with variable coefficients have been investigated using an established mathematical tool, the integral transform, together with a new analytic method developed in this study. This study aims to utilize the integral transform alongside the new analytical method. The new method was derived from the concept of exactness in higher-order ODEs. Specifically, second- and third-order ODEs with variable coefficients are exact if there exist first- and second-order linear ODEs whose derivatives correspond to the given equations, respectively. In this new analytic method, an integrating factor function formula for second-order ODEs has been carefully formulated and derived, making every second-order ODE with variable coefficients reducible to its lower-order form, specifically first-order ODEs. To ensure the accuracy of the new method, two well-known classes of second-order linear ODEs, namely the Whittaker second-order linear ODE and the Modified Bessel equation, were applied. The results demonstrated that the new analytic method effectively solves these equations, producing exact analytical solutions. To validate the effectiveness and efficiency of the new analytic method, a comparative analysis was conducted using illustrative examples, followed by graphical representations of the solution results.
VL  - 14
IS  - 2
ER  -

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