On the generalized solutions of a certain fourth order Euler equations
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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