    May  2021, 1(2): 127-140. doi: 10.3934/steme.2021010

## Unification of the common methods for solving the first-order linear ordinary differential equations

 1 School of Engineering and Technology, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4702, Australia

* Correspondence: w.guo@cqu.edu.au; Tel: +61-7-49309687

Received  April 2021 Revised  May 2021 Published  May 2021

A good understanding of the mathematical processes of solving the first-order linear ordinary differential equations (ODEs) is the foundation for undergraduate students in science and engineering programs to progress smoothly to advanced ODEs and/or partial differential equations (PDEs) later. However, different methods for solving the first-order linear ODEs are presented in various textbooks and resources, which often confuses students in their choice of the method for solving the ODEs. This special tutorial note presents the practices the author used to address such confusions in solving the first-order linear ODEs for students engaged in the bachelorette engineering studies at a regional university in Australia in recent years. The derivation processes of the four commonly adopted methods for solving the first-order linear ODEs, including three explicit methods and one implicit method presented in many textbooks, are presented first, followed by the logical interconnections that unify these four methods to clarify student's confusions on different presentations of the procedures and the solutions in different sources. Comparisons among these methods are also made.

Citation: William Guo. Unification of the common methods for solving the first-order linear ordinary differential equations. STEM Education, 2021, 1 (2) : 127-140. doi: 10.3934/steme.2021010
##### References:
  Greenberg, M.D., Advanced Engineering Mathematics. 2nd ed. 1998, Upper Saddle River, USA: Prentice Hall. Guo, W.W., Advanced Mathematics for Engineering and Applied Sciences. 2014, Sydney, Australia: Pearson. Bird, J., Higher Engineering Mathematics. 7th ed. 2014, UK: Routledge. Croft, A., Davison, R., Engineering Mathematics. 3rd ed. 2008, Harlow, UK: Pearson. James, G., Modern Engineering Mathematics. 2nd ed. 1996, Harlow, UK: Addison-Wesley Longman. Nagle, R.K., Saff, E.B., Fundamentals of Differential Equations. 3rd ed. 1993, USA: Addison-Wesley. Trim, D., Calculus for Engineers. 4th ed. 2008, Toronto, Canada: Pearson Stroud, K.A., Booth, D.J., Engineering Mathematics. 7th ed. 2013, London, UK: Palgrave McMillian. Zill, D.G., A First Course in Differential Equations with Modeling Applications. 10th ed. 2013, Boston, USA: Cengage Learning. Kreyszig, E., Advanced Engineering Mathematics. 10th ed. 2011, USA: Wiley. show all references

##### References:
  Greenberg, M.D., Advanced Engineering Mathematics. 2nd ed. 1998, Upper Saddle River, USA: Prentice Hall. Guo, W.W., Advanced Mathematics for Engineering and Applied Sciences. 2014, Sydney, Australia: Pearson. Bird, J., Higher Engineering Mathematics. 7th ed. 2014, UK: Routledge. Croft, A., Davison, R., Engineering Mathematics. 3rd ed. 2008, Harlow, UK: Pearson. James, G., Modern Engineering Mathematics. 2nd ed. 1996, Harlow, UK: Addison-Wesley Longman. Nagle, R.K., Saff, E.B., Fundamentals of Differential Equations. 3rd ed. 1993, USA: Addison-Wesley. Trim, D., Calculus for Engineers. 4th ed. 2008, Toronto, Canada: Pearson Stroud, K.A., Booth, D.J., Engineering Mathematics. 7th ed. 2013, London, UK: Palgrave McMillian. Zill, D.G., A First Course in Differential Equations with Modeling Applications. 10th ed. 2013, Boston, USA: Cengage Learning. Kreyszig, E., Advanced Engineering Mathematics. 10th ed. 2011, USA: Wiley. Iasson Karafyllis, Lars Grüne. 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