A Note on the Generalized Solutions of the Third-order Cauchy-Euler Equations

Authors

  • Nutgamol Sacorn Department of Mathematics, Khon Kaen University, Khon Kaen
  • Kamsing Nonlaopon Department of Mathematics, Khon Kaen University, Khon Kaen
  • Hwajoon Kim Faculty of General Education, Kyungdong University, Gyeonggi

DOI:

https://doi.org/10.26713/cma.v9i4.879

Keywords:

Generalized solutions, Distributional solutions, Weak solutions, Dirac delta function, Cauchy-Euler equation, Laplace transform

Abstract

In this paper, we propose the generalized solutions of the third order Cauchy-Euler equations at3y‴(t)+bt2y″(t)+cty′(t)+dy(t)=0, where a,b,c and d are real constants with a≠0 and t∈R using Laplace transform technique. We find that the types of solutions depend on the conditions of the values of a,b,c and d. Precisely, we obtain a distributional solution if (k3+3k2+2k)a−(k2+k)b+kc−d=0, for all k∈N and a weak solution if (k3−3k2+2k)a+(k2−k)b+kc+d=0, for all k∈N∪{0}. Our work improves the result of A. Kananthai [Distribution solutions of the third order Euler equation, Southeast Asian Bull. Math. 23 (1999), 627-631].

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References

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Published

25-12-2018
CITATION

How to Cite

Sacorn, N., Nonlaopon, K., & Kim, H. (2018). A Note on the Generalized Solutions of the Third-order Cauchy-Euler Equations. Communications in Mathematics and Applications, 9(4), 661–669. https://doi.org/10.26713/cma.v9i4.879

Issue

Vol. 9 No. 4 (2018)

Section

Research Article

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