<p>This study establishes an ordinary differential equation (ODE) model for continuous-time portfolio optimisation which addresses time-dependent risk aversion in mean-variance problems. The model includes practical constraints, including sector limits, a quadratic transaction cost penalty, and a total transaction cost budget. The Euler and fourth-order Runge-Kutta (RK4) methods were evaluated through numerical experiments to determine their accuracy, stability, and convergence properties. This study demonstrates how different numerical approaches impact portfolio performance, particularly when transaction costs are involved. The model undergoes a sensitivity analysis to evaluate its robustness by testing risk aversion levels and trading expenses. The model produces traditional mean-variance results when costs are absent, but generates stable adaptive portfolio adjustments when costs are included. The ODE-based method provides an efficient method for dynamically managing portfolios under actual trading restrictions.</p>