We derive simple analytical approximations (in explicit form) for the hydrodynamic evolution of the electron temperature T(s, t) and electron density n(s, t), for one-dimensional coronal loops that are subject to impulsive heating with subsequent cooling. Our analytical approximations are derived from first principles, using (1) the hydrodynamic energy balance equation, (2) the loop scaling laws of Rosner-Tucker-Vaiana and Serio, (3) the Neupert effect, and (4) the Jakimiec relationship. We compare our analytical approximations with 56 numerical cases of time-dependent hydrodynamic simulations from a parametric study of Tsiklauri et al., covering a large parameter space of heating rates, heating timescales, heating scale heights, loop lengths, for both footpoint and apex heating, mostly applicable to flare conditions. The average deviations from the average temperature and density values are typically 20% for our analytical expressions. The analytical approximations in explicit form provide an efficient tool to mimic time-dependent hydrodynamic simulations, to model observed soft X-rays and extreme-ultraviolet light curves of heated and cooling loops in the solar corona and in flares by forward fitting, to model microflares, to infer the coronal heating function from light curves of multi-wavelength observations, and to provide physical models of differential emission measure distributions for solar and stellar flares, coronae, and irradiance.