This paper systematically establishes a unified theory for solving differential equations based on the concept of differential algebraic closure, extending the framework previously developed for polynomial equations to the realm of differential equations. For a given differential equation F = 0, we construct its universal differential algebraic closure KeF and prove that for linear Fuchsian ODEs with algebraic function coefficients, secondorder nonlinear equations possessing the Painlev´e property, and finite-genus integrable systems, the solutions can be expressed as closed-form expressions obtained from the coefficients by a finite number of DE-operations (arithmetic operations, differentiation, elementary transcendental functions, θfunctions, and inversion). Main contributions include: 1. A unified solution formula for n-th order linear ODEs, expressing solutions as n-th roots of differential-algebraic parameters Φm(x) combined with a Fourier sum governed by uniformization indices σm(x) ∈Z/nZ. The equivalence with classical special function solutions (hypergeometric, Bessel) is proved. 2. A rigorous definition of the analytic genus g as the geometric genus of the spectral curve, together with a Riemann-Hurwitz type genus formula g =12(n− 1)(r − 2) −Pri=1n(di−1)2di, establishing a precise correspondence between singular point structure and solution complexity.3. For Fuchsian equations whose spectral curve is hyperelliptic of genus g ≥ 1, we construct the monodromic period matrix T (x) and prove a Theta-Thomae formula, thereby establishing an explicit differential isomorphism between the differential-algebraic solution field and the field of g-dimensional Riemann θ-functions. 4. For Painlev´e equations, via the isomonodromic deformation theory of Lax pairs, we prove that all Painlev´e transcendents belong to the DE class (level L4) and provide explicit Jacobi/Riemann θ-function representations for PII and PVI. 5. For Lax-integrable PDEs (KdV, NLS, sine-Gordon), we construct the universal differential algebraic closure Ke(g) KdV of genus g and prove a unified θ-function formula for finite- ap quasi-periodic solutions, together with explicit loop-integral algorithms for the period matrix, wavenumbers and frequencies.6. A modified Kovacic-van Hoeij algorithm for linear ODEs and a Painlev´eLax decision algorithm for nonlinear ODEs are presented, with complexity analysis. 7. A five-level hierarchy L1-L5 of finite representability is introduced and the strict inclusions are proved, clarifying the boundary of ”explicit solvability”. 8. A dimension formula dimM = 3g − 3 + ν for the moduli space of differential equations is derived, where ν counts isomonodromic parameters, with explicit verifications for Fuchsian systems and Painlev´e equations. The framework unifies Galois theory, differential Galois theory, isomonodromic deformation theory and θ-function theory, providing a new paradigm for problems in mathematical physics and laying foundations for p-adic differential equations and arithmetic dynamics.
shifa liu (Wed,) studied this question.