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Abstract
This paper extends the differential algebraic framework for solving polynomial equations to the domain of differential equations. We establish that solutions to linear ordinary differential equations of arbitrary order can be analytically expressed within a differential algebraic closure KDE, which extends the coefficient field with derivative operators, their
evaluations at critical points, and appropriate special functions. We provide complete constructive proofs, derive combinatorial expressions for the correction coefficients γ(n)
m,j, and present a detailed O(n2) algorithm. Extensive numerical validation demonstrates machine-precision accuracy (residuals <10−30) for various differential equations including Airy, Bessel, and Legendre equations. This work demonstrates that while closed-form solutions in elementary functions are impossible for many differential equations, explicit analytic solutions exist in the appropriately extended differential algebraic closure KDE.
Supplementary materials
Title
Extension of the Differential Algebraic Framework to Nonlinear Differential Equations: A Constructive Approach to Unified Analytic Solutions
Description
This paper establishes a constructive differential algebraic framework for obtaining explicit analytic solutions to broad classes of nonlinear ordinary differential equations (ODEs). We define the nonlinear differential algebraic closure KNLDE, a differentially closed field extension constructed through a recursive adjunction process that incorporates solutions to linearized equations, radical extensions, roots of unity, nonlinear special functions, and limits of approximating sequences. We provide complete constructive proofs, deriving explicit combinatorial expressions for the nonlinear correction coefficients Γ(n) m,k from first principles using the method of undetermined coefficients and asymptotic balancing. Detailed algorithms with complexity analysis are presented, alongside extensive numerical validation demonstrating machine-precision accuracy (residuals < 10−28 using high-precision arithmetic) for various nonlinear ODEs, including Riccati, Duffing, and Painlev´e equations.
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