**Volume 10, Issue 8, pp 4077--4084**

**Publication Date**: 2017-08-06

http://dx.doi.org/10.22436/jnsa.010.08.04

Amphon Liangprom - Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand.

Kamsing Nonlaopon - Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand.

In this paper, using Laplace transform technique, we propose the generalized solutions of the fourth order Euler differential equations \[t^4y^{(4)}(t)+t^3y'''(t)+t^2y''(t)+ty'(t)+my(t)=0,\] where \(m\) is an integer and \(t\in\mathbb{R}\). We find types of solutions depend on the values of \(m\). Precisely, we have a distributional solution for \(m=-k^4-5k^3-9k^2-4k\) and a weak solution for \(m=-k^4+5k^3-9k^2+4k,\) where \(k\in\mathbb{N}.\)

Generalized solution, distributional solution, Euler equation, Dirac delta function.

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