As quantum hardware rapidly advances toward the early fault-tolerant era, a key challenge is to develop quantum algorithms that are not only theoretically sound but also hardware-friendly on near-term devices. In this work, we propose a quantum algorithm for solving linear ordinary differential equations (ODEs) with a provable runtime guarantee. Our algorithm uses only a single ancilla qubit and is locality preserving: when the coefficient matrix of the ODE is k-local, the algorithm only needs to implement the time evolution of (k + 1)-local Hamiltonians. We also discuss the connection between our proposed algorithm and Lindbladian simulation. By applying our algorithm to the interacting Hatano–Nelson model, a widely studied non-Hermitian model with rich phenomenology, and numerically simulating it under realistic noise models, we demonstrate its practical feasibility on near-term quantum devices.