Time-varying virtual analog filters used in digital audio effects and synthesizers are often implemented by discretizing continuous-time state-space systems using trapezoidal integration. When filter parameters such as cutoff frequency or resonance vary over time, as is the norm in musical applications, proving BIBO stability of the resulting time-varying discrete-time system becomes nontrivial. In this paper, we review the technique of common quadratic Lyapunov functions (CQLFs) from the control systems literature and show how a continuous-time CQLF is preserved through discretization. This allows us to prove stability of some time-varying virtual analog filters by working in the often simpler continuous-time domain. We apply this framework to several filters of musical interest, providing new proofs of stability for the state variable filter and the Sallen-Key filter, and new bounds on the stable time-varying parameter range for the Moog ladder and diode ladder filters.