This paper examines the existence and uniqueness of the solutions of second-order neutrosophic differential equations by assuming a condition of Lipschitz continuity. The discussion extends to the general case of nth-order neutrosophic differential equations with initial value conditions. The theoretical framework used in this research is founded on the Banach fixed-point theorem, which guarantees the existence and uniqueness of solutions under a neutrosophic setting. Such a generalization provides a remarkable improvement in the modeling of dynamic systems that take into account the aspects of truth, indeterminacy, and falsity simultaneously. To confirm the theoretical findings, a number of demonstrative examples are given, proving the efficiency and applicability of the suggested approach in solving intricate differential systems marked by uncertainty and inconsistency.