Final answer:
The true initial conditions of the nonlinear spring system described by the differential equation are unclear due to a discrepancy in the provided information. Without solving the equation or additional details, the long-term behavior of the system, such as whether x approaches infinity as t approaches infinity, cannot be determined.
Step-by-step explanation:
The differential equation provided in the question describes a nonlinear spring system, indicating it's a physics problem typically studied at the college level involving complex dynamics. Given that the system described is non-linear based on the inclusion of the terms ex^3 and e2x^5, any behavior of the system would depend on the detailed analysis or numerical solution of the equation, which is not provided here. Moreover, the initial condition given in the question, x(0) = -5, contradicts the provided information stating x(0) = 0, resulting in a discrepancy regarding the true initial conditions.
The claim about the long-term behavior of the system, x → [infinity] as t → [infinity], cannot be confirmed or refuted without solving the differential equation or more details about the system's potential energy and damping characteristics. In the theory of nonlinear dynamics, long-term behavior can indeed be complex, and one cannot readily assume it approaches infinity without proper evidence.
Newton's third law applies to the external forces acting on a system and would indeed influence the behavior of a spring system, though this doesn't directly pertain to the focal question of the long-term behavior of x(t).