  We obtain exact results for the recently discovered finite-time thermodynamic
uncertainty relation in a stochastically driven system with non-Gaussian work
statistics, both in the steady state and transient regimes, by obtaining exact
expressions for any moment of the dissipated work at arbitrary times. The
uncertainty function (the Fano factor of the dissipated work) is bounded from
below by $2k_BT$ as expected, for all times $\tau$, in both steady state and
transient regimes. The lower bound is reached at $\tau=0$ as well as when
certain system parameters vanish (corresponding to an equilibrium state).
Surprisingly, we find that the uncertainty function also reaches a constant
value at large $\tau$ for all the cases we have looked at. For a system
starting and remaining in steady state, the uncertainty function increases
monotonically, as a function of $\tau$ as well as other system parameters,
implying that the large $\tau$ value is also an upper bound. For the same
system in the transient regime, however, we find that the uncertainty function
can have a local minimum at an accessible time $\tau_m$, for a range of
parameter values. The non-monotonicity suggests, rather counter-intuitively,
that there might be an optimal time for the working of microscopic machines, as
well as an optimal configuration in the phase space of parameter values. Our
solutions show that the ratios of higher moments of the dissipated work are
also bounded from below by $2k_BT$. For another model, also solvable by our
methods, which never reaches a steady state, the uncertainty function, is in
some cases, bounded from below by a value less than $2k_BT$.
