The Generalized Solutions of Second and Third-Order  Cauchy-Euler  Equations by Using the Elzaki Transforms

Piyatida    Chanapun; Karuna   Kaewnimit

Authors

Piyatida Chanapun Department of Mathematics, Faculty of Education, Roi Et Rajabhat University
Karuna Kaewnimit Department of Mathematics, Faculty of Education, Roi Et Rajabhat University

Keywords:

Cauchy-Euler equation, Dirac delta function , Elzaki transform, The generalized solutions

Abstract

This paper aims to study the generalized solutions of Cauchy-Euler equations of the form $t^{2}y^{"}(t)+aty'(t)+by(t)=0$ , and $t^{3}y'''(t)+at^{2}y''(t)+bty'(t)+cy(t)=0,$ where a, b, and c are integers and $t\in R,$ using Elzaki transform technique. The solutions are in the space of distributions. Types of solutions are in the form of a distributional solution $y(t)=\sigma^{k}(t)$ and a weak solution $y(t)=\frac{t^{k}H(t)}{k!}$ which depends on the values of a, b, and c.

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