This paper establishes a complete unified framework that organically links matrices, differential equations, and integral equations. We systematically study and elaborate in detail the Schr\"odinger equation, Pauli equation, Klein-Gordon equation, and Dirac equation in quantum mechanics; the Maxwell equation, Yang-Mills equation, Schwinger-Dyson equation, Bethe-Salpeter equation, and renormalization group equation in quantum field theory; the Einstein field equation in general relativity; the Friedmann equation in cosmology; the Gell-Mann matrix equation, CKM matrix equation, and PMNS matrix equation in particle physics; the supersymmetric quantum mechanics equation and supersymmetric Yang-Mills equation in supersymmetry; the bosonic string equation and superstring equation in string theory; the Ginzburg-Landau equation and Bogoliubov-de Gennes equation in condensed matter physics; the BCS theory of superconductivity; and the Boltzmann equation, Langevin equation, and Fokker-Planck equation in nonequilibrium statistical physics. For each equation, we provide: matrix forms, eigenvalue equations, first-order, second-order, third-order, and arbitrary higher-order differential equations (with and without fundamental constants), first-order, second-order, third-order, and arbitrary higher-order integral equations (with and without fundamental constants), and stochastic generalizations. Based on this, we establish unified first-order and higher-order differential equations, unified first-order and higher-order integral equations, and a unified stochastic differential equation framework, and prove that all matrices and their corresponding equations can be derived from this unified framework. All theorems are given with rigorous proofs, and all derivations contain complete steps. Infinite-dimensional operators are treated rigorously using spectral theory, semigroup theory, and the Hille-Yosida theorem. This system, which combines unification with specificity, reveals the profound mathematical connections among equations from different physical backgrounds, providing a unified mathematical description for theoretical physics.
S. B. Liu (Wed,) studied this question.