This technical note considers the switching time optimization of time-varying linear switched systems subject to quadratic cost-also potentially time-varying. The problem is formulated so that only a single set of differential equations need to be solved prior to optimization. Once these differential equations have been solved, the cost may be minimized over an arbitrary number of modes and mode sequences without requiring additional simulation. The differential equations that need to be simulated are as smooth as the system's vector fields, despite the fact that the optimization itself is nonsmooth. The technique is illustrated in examples from the literature.