  Detecting the environment around the supermassive black holes and tests of
general relativity are important applications of extreme-mass-ratio inspirals
(EMRIs). There is still a challenge to efficiently describe various "dirty"
impacts on the inspirals like dark matter, gas, dipole radiation,
electromagnetic interaction, and so on. In this Letter, we find the inherent
linearity of the asymptotic solution of the inhomogeneous Teukolsky equation.
Based on this property, we completely decouple the factors of the perturber and
the background spacetime in the energy fluxes and waveforms. With the new
decoupling form, the waveforms of EMRIs with non-geodesic motion in Kerr
spacetime can be calculated conveniently. This will help to resolve the
environment (including gas, field, dark matter, electromagnetic interaction,
etc.) around the supermassive black holes and test general relativity.
