7(b)

The polynomial $.5 D^2 + 8 D + 32 = .5 (D+8)^2$ has a double root $-8$ (it must be a double root, of course, for the system to be critically damped). Thus the general solution is $y = (c_1 + c_2 t) \exp(-8t)$. From the initial conditions (with down negative), $y(0) = c_1 = 1/4$ and, since $y' = (-8 c_1 + c_2 - 8 c_2 t) \exp(-8t)$, $y'(0) = -8 c_1 + c_2 = -5$, so $c_2 = -3$. The solution is $y = (1/4 - 3 t) \exp(-8t)$, which crosses the equilibrium point at $t = 1/12$ second.