On the generalized solutions of a certain fourth order Euler equations

Volume 10, Issue 8, pp 4077--4084 Publication Date: August 06, 2017

Authors

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

Abstract

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 $k\in\mathbb{N}.$

Keywords

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

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