  We reexamine the general solution of a Schr\"{o}dinger equation in the
presence of a time-dependent linear potential in configuration space based on
the Lewis-Riesenfeld framework. For comparison, we also solve the problem in
momentum space and then Fourier transform the solution to get the general wave
function. Appropriately choosing the weight function in the latter method, we
can obtain the same wave function as the former method. It is found that
non-Hermitian time-dependent linear invariant can be used to obtain
Gaussian-type wave-packet (GTWP) solutions of the time-dependent system. This
operator is a specific linear combination of the initial momentum and initial
position operators. This fact indicates that the constants of integration such
as the initial position and initial momentum that determine the classical
motion play important roles in the time-dependent quantum system. The
eigenfunction of the linear invariant is interpreted as a wave packet with a
"center of mass" moving along the classical trajectory, while the ratio between
the coefficients of the initial position and initial momentum determines the
width of the wave packet.
