Exact analytical approach to differential equations with variable coefficients

TL;DR

This paper presents a formal analytical method for solving differential equations of any order with variable coefficients, demonstrating how approximate solutions can be derived from the exact expression, with potential applications in quantum mechanics.

Contribution

The paper introduces a novel formal analytical approach for arbitrary order variable coefficient differential equations, extending to infinite-dimensional cases like quantum Hamiltonians.

Findings

Exact analytical solutions can be constructed for variable coefficient differential equations.

Approximate solutions are derivable from the analytical expressions.

Method can be extended to infinite-dimensional problems such as quantum Hamiltonians.

Abstract

This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the analytical expression presented in the paper. The formalism can be easily extended to the infinite dimensional case such as the quantum time-dependent Hamiltonian problem.