An innocent linear test decides the fate of every explicit integrator. Write y′ = λy with Re(λ) ≪ 0. While |λ| stays modest a fifth-order Runge–Kutta pair advances with large, inexpensive steps; the moment |λ| grows the stability region collapses and those same steps shrink until the computation is useless. Stiffness is that mismatch of time scales. The solution itself may already be smooth, yet an explicit method is still forced to resolve the fastest decaying transient. Implicit formulae (BDF, Radau collocation, Rosenbrock) replace the explicit update with a linear or nonlinear solve and remain stable on the slow manifold. Curtiss and Hirschfelder first called such equations “stiff” in 1952; the same test still separates the solvers that finish from those that do not.