  We present a technique, {\em the uniform asymptotic approximation}, to
construct accurate analytical solutions of the linear perturbations of
inflation after quantum effects of the early universe are taken into account,
for which the dispersion relations generically become nonlinear. We construct
explicitly the error bounds associated with the approximations and then study
them in detail. With the understanding of the errors and the proper choice of
the Liouville transformations of the differential equations of the
perturbations, we show that the analytical solutions describe the exact
evolution of the linear perturbations extremely well even only in the
first-order approximations. As an application of the approximate analytical
solutions, we calculate the power spectra and indices of scalar and tensor
perturbations in the slow-roll inflation, and find that the amplitudes of the
power spectra get modified due to the quantum effects, while the power spectrum
indices remain the same as in the linear case.
