The solution to the initial value problem

\begin{eqnarray*} \frac{d^2x}{dt^2} + \frac 12 \frac{dx}{dt} + 4x & = & \cos(\omega t) \\ \\ x(0) & = & 0 \\ \frac{dx}{dt}(0) & = & 0 \\ \end{eqnarray*}