Explicit Analytic Solutions to Total Differential Equations via Constructive Differential Algebraic Closure

This paper establishes a comprehensive constructive framework for obtaining explicit analytic solutions to linear total differential equations. We introduce the concept of total differential algebraic closure KTDE, a finitely generated extension of the coefficient field that contains all necessary elements for expressing solutions to a broad class of total differential equations. The closure is constructed through a systematic tower of extensions incorporating coefficients, boundary values, fundamental solutions, and specific algebraic elements. The solution formula features meticulously derived combinatorial correction terms γ(n) m based on Stirling numbers of the second kind, with rigorous foundation in the multivariate Fa`a di Bruno formula adapted for total differential operators. We provide complete existence proofs using the analytic implicit function theorem and develop a comprehensive algorithmic framework with detailed complexity analysis. Numerical experiments demonstrate spectral convergence for analytic solutions and confirm the necessity of combinatorial corrections for higher-order equations. The framework is carefully reconciled with classical impossibility results, showing that solutions reside in properly constructed extensions beyond elementary functions. Connections to differential Galois theory, exterior differential systems, and modern solvability theory are established.