The generalized solutions of a certain n order differential equations with polynomial coefficients

Abstract

In this paper, we propose the generalized solutions of the differential equation $t{y}^{\left(n\right)}\left(t\right)+m{y}^{(n-1)}\left(t\right)+ty\left(t\right)=0,$ where m and n are any integers with $n\ge 2$, and $t\in \mathbb{R}$ using Laplace transform technique. We find that the types of Laplace transformable solutions in the space of right-sided distributions depend on the relationship between m and n. Precisely, we have a distributional solution provided $m\ge n$, and a weak solution otherwise. Our work improves the result of Kananthai [The distribution solutions of ordinary differential equation with polynomial coefficients. Southeast Asian Bull Math. 2001;25:129–134].

Keywords:

• Dirac delta function
• generalized solution
• distributional solution
• Laplace transform

AMS Subject Classification:

• 34A37
• 34B30
• 44A10
• 46F10
• 46F12

Disclosure statement

No potential conflict of interest was reported by the authors.