A Note on the Generalized Solutions of the Third-order Cauchy-Euler Equations - Sacorn, Nutgamol; Nonlaopon, Kamsing; Kim, Hwajoon (Communications in Mathematics and Applications, 2018)
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 \(a eq 0\) and \(tR\) 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 \(k N\) and a weak solution if \((k^3 - 3k^2 + 2k)a + (k^2 - k)b + kc + d = 0\), for all \(k N{0}\). Our work improves the result of \(A\). Kananthai [Distribution solutions of the third order Euler equation, Southeast Asian Bull. Math. 23 (1999), 627-631].