Distribution solutions of the third order Euler equation (Q1964666)

Consider the third-order Euler differential equation of the form \[ t^3 y'''(t)+ t^2 y''(t)+ ty'(t)+ my(t)= 0, \] where \(m\in\mathbb{Z}\) and \(t\in\mathbb{R}\). In this paper using the theory of Laplace transform of right sided distributions, the author proves that if \(m= -k^3+ 2k^2- 2k\) \((k\in\mathbb{N})\) then the above equation admits continuous solution whereas if \(m= k^3+ 2k^2+ 2k\) \((k\in\mathbb{N})\) then the above equation admits solutions in the space of distributions. In fact the classical solutions are \(H(t)t^k/k!\) for \(m= -k^3+ 2k^2- 2k\) and \(\delta^{k-1}\) for \(m= k^3+ 2k^2+ 2k\) where \(H(t)\) is the Heaviside function and \(\delta^{k-1}\) denotes the distributional \((k-1)\)th derivative of the Dirac delta functional \(\delta\).