The recently developed differential algebraic closure framework for polynomial equations 1 provides explicit finite closed-form expressions for all roots of any degree-n polynomial using only arithmetic operations, differentiation, evaluation, and n-th root extraction. In this paper, we show that this framework can be faithfully and correctly extended to a large class of linear ordinary differential equations by applying it not to the solution functions themselves, but to the characteristic polynomial (or its appropriate algebraic counterpart). For constant-coefficient linear ODEs, the characteristic polynomial is a polynomial with constant coefficients; its roots (eigenvalues) are algebraic numbers whenever the coefficients are rational numbers. Applying the polynomial differential algebraic closure theorem 1 to this characteristic polynomial yields an explicit DAODE expression for each eigenvalue. The general solution is then a linear combination of exponentials of these eigenvalues — a formula that belongs to the extended complexity class DAODEexp if the exponentials themselves are treated as primitive constants, or simply to DAODE when the exponentials are evaluated at algebraic arguments. For Euler–Cauchy equations, the indicial polynomial plays the same rôle; its roots are again obtained via the polynomial closure. For Fuchsian linear ODEs with finite monodromy group, all solutions are algebraic functions. The minimal polynomial of such an algebraic function can be constructed from the monodromy representation; once this polynomial is obtained, the differential algebraic closure theorem directly supplies a finite DAODE expression for the solutions. For integrable nonlinear ODEs (e.g., Painlevé equations) that admit a Lax pair representation, the spectral curve is an algebraic curve; its equation is a polynomial in the spectral parameter whose coefficients are constants of motion. The roots of this polynomial — the algebraic eigenvalues of the Lax matrix — are again accessible via the polynomial closure. The theta-functional solutions of the original nonlinear equation are then expressed rationally in terms of these roots and their N-th roots, establishing membership in DAODE (or its natural extension). Thus, every linear ODE with constant or Euler type, every Fuchsian ODE with finite monodromy, and every algebraically integrabl nonlinear ODE admits a finite DAODE representation of its solutions, obtained by applying the polynomial differential algebraic closure to a single associated polynomial — the characteristic polynomial, the indicial polynomial, the minimal polynomial of the algebraic solution, or the spectral polynomial. This completely avoids the erroneous direct generalisation attempted in earlier works and provides a mathematically rigorous, fully constructive unification. We furthermore prove a Coefficient Finite Representation Theorem: all coefficients of the associated polynomial, as well as all branch indicators arising from the root extraction, are themselves DAODE functions of the original differential equation’s coefficients. This reveals the extraordinary datacompression power of the differential algebraic closure language. A precise characterisation of DAODE-finitely representable transcendental functions is given: a function is DAODE-finitely representable iff it is an algebraic function (satisfies a polynomial equation with coefficients in the base field). This characterisation, together with the classical theory of algebraic functions and their monodromy, yields a complete classification. Functions such as Ai(x), Bi(x), Jν(x) with irrational ν, Γ(x), ζ(x) are not DAODEfinitely representable; they belong to higher levels of the representability hierarchy introduced herein. Applications are given to: (i) fast certified computation of eigenvalues of constant-coefficient ODEs via polynomial closure; (ii) explicit algebraic expressions for Legendre polynomials and other classical orthogonal polynomials; (iii) hypergeometric functions with algebraic parameters; (iv) the Yablonskii–Vorob’ev polynomial solutions of the second Painlevé equation; (v) finite-gap solutions of the KdV equation and their reduction to solitons. All algorithms are implemented in the SageMath package daclosure and numerical verification is provided. This work repositions the differential algebraic closure method as a tool for solving polynomial equations, and demonstrates how it becomes a powerful weapon for differential equations once the latter are reduced to their characteristic or spectral polynomials.
shifa liu (Wed,) studied this question.