6(a)

The differential equation is $y'' + 2 y' + 2 y = 0$. If down is negative, the initial conditions are $y(0) = 0$, $y'(0) = -5$. The polynomial $D^2 + 2 D + 2$ has roots $-1 \pm i$, so the general solution is $y = \exp(-t)( c_1 \cos t + c_2 \sin t)$. From the initial conditions, $y(0) = 0 = c_1$ and, since $y' = \exp(-t)((-c_1 + c_2) \cos t + (-c_1 - c_2) \sin t)$, $y'(0) = -c_1 + c_2 = 5$, so $c_2 = -5$ and $y = -5 \exp(-t) \sin t$.