The numerical solution of time-dependent differential equations constitutes a fundamental pillar of computational applied mathematics, enabling accurate simulation of dynamic processes in engineering, physics, and data-driven modeling applications. This comprehensive study examines stability analysis and error estimation techniques for numerical methods applied to time-dependent systems, with emphasis on mathematical rigor, computational efficiency, and practical implementation considerations. The investigation encompasses classical approaches including finite difference time-stepping schemes, finite element temporal discretizations, and spectral methods, alongside emerging techniques integrating machine learning with traditional numerical frameworks. Particular attention is devoted to stability theory including absolute stability regions, stiffness considerations, and the Courant-Friedrichs-Lewy condition, as well as convergence analysis establishing relationships between temporal and spatial discretization errors. Error estimation methodologies including local truncation error analysis, global error propagation, and adaptive time-stepping strategies are systematically examined to provide practical guidance for achieving desired accuracy levels while maintaining computational efficiency. Application domains explored include transient heat conduction, wave propagation phenomena, structural dynamics, computational fluid dynamics, and modern data assimilation systems requiring real-time numerical solution capabilities. The analysis reveals that optimal numerical method selection depends critically on problem characteristics including stiffness, smoothness, time scale separation, and computational budget constraints. This work provides researchers and practitioners with systematic frameworks for analyzing, selecting, and implementing time integration schemes appropriate for diverse application requirements, while identifying critical challenges and promising research directions in numerical analysis of time-dependent systems.