Suppose $L=20$ henrys, $R=40$ ohms, $C=\frac{1}{300}$ farads, $E=100$ volts, $q(0)=4$ coulombs, and $q^{\text{'}}(0)=I(0)=0.$ Formulate and solve an initial value problem that models the given $LRC$ circuit.

$q(t)=$

$($Type an exact answer, using radicals as needed.$)$