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Inspiring and engaging high school students with science and technology education in regional Australia
Unification of the common methods for solving the firstorder linear ordinary differential equations
1.  School of Engineering and Technology, Central Queensland University, Bruce Highway, North Rockhampton, QLD 4702, Australia 
A good understanding of the mathematical processes of solving the firstorder 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 firstorder 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 firstorder 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 firstorder 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.
References:
[1] 
Greenberg, M.D., Advanced Engineering Mathematics. 2nd ed. 1998, Upper Saddle River, USA: Prentice Hall. Google Scholar 
[2] 
Guo, W.W., Advanced Mathematics for Engineering and Applied Sciences. 2014, Sydney, Australia: Pearson. Google Scholar 
[3] 
Bird, J., Higher Engineering Mathematics. 7th ed. 2014, UK: Routledge. Google Scholar 
[4] 
Croft, A., Davison, R., Engineering Mathematics. 3rd ed. 2008, Harlow, UK: Pearson. Google Scholar 
[5] 
James, G., Modern Engineering Mathematics. 2nd ed. 1996, Harlow, UK: AddisonWesley Longman. Google Scholar 
[6] 
Nagle, R.K., Saff, E.B., Fundamentals of Differential Equations. 3rd ed. 1993, USA: AddisonWesley. Google Scholar 
[7] 
Trim, D., Calculus for Engineers. 4th ed. 2008, Toronto, Canada: Pearson Google Scholar 
[8] 
Stroud, K.A., Booth, D.J., Engineering Mathematics. 7th ed. 2013, London, UK: Palgrave McMillian. Google Scholar 
[9] 
Zill, D.G., A First Course in Differential Equations with Modeling Applications. 10th ed. 2013, Boston, USA: Cengage Learning. Google Scholar 
[10] 
Kreyszig, E., Advanced Engineering Mathematics. 10th ed. 2011, USA: Wiley. Google Scholar 
show all references
References:
[1] 
Greenberg, M.D., Advanced Engineering Mathematics. 2nd ed. 1998, Upper Saddle River, USA: Prentice Hall. Google Scholar 
[2] 
Guo, W.W., Advanced Mathematics for Engineering and Applied Sciences. 2014, Sydney, Australia: Pearson. Google Scholar 
[3] 
Bird, J., Higher Engineering Mathematics. 7th ed. 2014, UK: Routledge. Google Scholar 
[4] 
Croft, A., Davison, R., Engineering Mathematics. 3rd ed. 2008, Harlow, UK: Pearson. Google Scholar 
[5] 
James, G., Modern Engineering Mathematics. 2nd ed. 1996, Harlow, UK: AddisonWesley Longman. Google Scholar 
[6] 
Nagle, R.K., Saff, E.B., Fundamentals of Differential Equations. 3rd ed. 1993, USA: AddisonWesley. Google Scholar 
[7] 
Trim, D., Calculus for Engineers. 4th ed. 2008, Toronto, Canada: Pearson Google Scholar 
[8] 
Stroud, K.A., Booth, D.J., Engineering Mathematics. 7th ed. 2013, London, UK: Palgrave McMillian. Google Scholar 
[9] 
Zill, D.G., A First Course in Differential Equations with Modeling Applications. 10th ed. 2013, Boston, USA: Cengage Learning. Google Scholar 
[10] 
Kreyszig, E., Advanced Engineering Mathematics. 10th ed. 2011, USA: Wiley. Google Scholar 
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