Structural Limits of Closed-Form Solutions for Second-Order Differential Equations under Admissibility

Description (Abstract):

This note establishes a definitive structural boundary on the existence of universal closed-form solutions for second-order ordinary differential equations (ODEs). Moving beyond questions of symbolic solvability within specific function classes, the analysis applies an admissibility framework that rigorously distinguishes between structurally viable and non-viable solution claims. It is demonstrated that no universal finite closed-form solution exists for the general class of second-order ODEs under any fixed symbolic function system. The strongest admissible universal representation is shown to be an operator-level construct: ordered exponentials (or fundamental matrix formulations) in the linear case, and flow-map definitions in the nonlinear case. This result reframes a longstanding pursuit not as an unresolved technical challenge, but as a question of inherent structural possibility, thereby clarifying the ultimate limit of symbolic compression permitted by mathematics for this class of problems.