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

Abstract

In this paper, we propose the generalized solutions of the third order Cauchy-Euler equations $$at^3y'''(t) + bt^2y''(t) + cty'(t) + dy(t)=0,$$ where \(a, b, c\) and \(d\) are real constants with \(a \neq 0\) and \(t\in\mathbb{R}\) 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 \((k^3 + 3k^2 + 2k)a - (k^2 + k)b + kc - d = 0\), for all \(k \in \mathbb{N}\) and a weak solution if \((k^3 - 3k^2 + 2k)a + (k^2 - k)b + kc + d = 0\), for all \(k \in \mathbb{N}\cup\{0\}\). Our work improves the result of \(A\). Kananthai [Distribution solutions of the third order Euler equation, Southeast Asian Bull. Math. 23 (1999), 627-631].