Backward Euler, unlike forward Euler, offers unconditional stability for ODEs, even with large time steps. This is because its stability condition (1 + ah > 1 or 1 + ah < -1) is always met for positive time steps. However, large time steps don't guarantee accuracy; stability doesn't imply accuracy. Backward Euler requires solving a linear system per iteration (more computationally expensive than forward Euler's simple matrix multiplication), but allows for fewer, larger, stable time steps. This segment discusses the crucial trade-off between stability and accuracy when using numerical methods to solve ODEs. The speaker emphasizes that while backward Euler guarantees stability even with large time steps, this doesn't imply accuracy. Using a hypothetical example of an extremely large time step, the speaker illustrates how a stable solution might be completely unrealistic, highlighting the importance of choosing appropriate time steps for both stability and accuracy in numerical simulations. This segment details a step-by-step stability analysis of the backward Euler method for solving ordinary differential equations (ODEs). The speaker meticulously examines the conditions under which the method remains stable, highlighting the importance of the magnitude of a constant related to the time step and the ODE's coefficient. Despite initial errors in the calculations, the speaker corrects the mistakes and arrives at the correct stability condition, demonstrating a rigorous approach to numerical analysis.