A new transform for solving linear second-orders ODE with variable coefficients

Abstract

In this paper, we present a novel symmetry-enhanced transform
to evaluate Gaussian integrals commonly used in mathematical and
physical domains, particularly in quantum field theory. Additionally,
we utilize this original transformation methodology to solve a wide
range of second-order linear ordinary differential equations (ODEs)
that have variable coefficients, which is a common occurrence in physics.
Notable examples encompass Weber, Euler-Cauchy, and Bessel equa-
tions, highlighting the broad applicability of our proposed method.
Diverging from established transforms like the widely used Laplace
transform, our innovative approach introduces a symmetrical model,
offering a distinct and founding perspective to the field.

Keywords: New integral transform, Gaussian integral, Equation of free oscillations, Weber’s equation, Euler-Cauchy Equation, Bessel’s equation.