One-dimensional system of Brownian motions called Dyson's model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is beta/2 times the inverse of particle distance. When beta = 2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson's model with beta = 2 and N particles, X(t) = (X(1)(t), ... , X(N)(t)), t. [0, infinity), 2 <= N <= infinity, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function Ai(z) is an entire function with zeros all located on the negative part of the real axis R. We consider Dyson's model with beta = 2 starting from the first N zeros of Ai(z), 0 > a(1) > ... > a(N), N >= 2. In order to properly control the effect of such initial confinement of particles in the negative region of R, we put the drift term to each Brownian motion, which increases in time as a parabolic function: Y(j) (t) = X(j) (t) + t(2)/4 + {d(1) + Sigma(N)(l=1)(1/al)}t, 1 <= j <= N, where d(1) = Ai'(0)/Ai(0). We show that, as the N -> infinity limit of Y(t) = (Y(1)(t), ..., Y(N)(t)), t is an element of [0, infinity), we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of Ai(z) on the negative R is occupied by one particle, to the stationary state mu(Ai). The stationary state mu(Ai) is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on R and in which the Tracy-Widom distribution describes the rightmost particle position.