  The non-equilibrium dynamics of electrons is of a great experimental and
theoretical value providing important microscopic parameters of the Coulomb and
electron-phonon interactions in metals and other cold plasmas. Because of the
mathematical complexity of collision integrals theories of electron relaxation
often rely on the assumption that electrons are in a "quasi-equilibrium" (QE)
with a time-dependent temperature, or on the numerical integration of the
time-dependent Boltzmann equation. We transform the integral Boltzmann equation
to a partial differential Schroedinger-like equation with imaginary time in a
one-dimensional "coordinate" space reciprocal to energy which allows for exact
analytical solutions in both cases of electron-electron and electron-phonon
relaxation. The exact relaxation rates are compared with the QE relaxation
rates at high and low temperatures.
