How was Vc(t) found in this source free RC circuit?
When \$t &lt; 0\$ (Which is not the same as "after \$t=0\$"), then yes, \$10^4\space i_s = v_c(t)\$. That's exactly what the final line of working out says. (Hint: what is \$2/3 \mathrm{mA} \times 10^4\$?). 

The implication of \$t=0\$ above the switch is that the switch is closed from \$t=-\infty\$ to \$t=0\$, so it is in a steady state condition just before the switch is opened. In steady state the capacitor is fully charged and all the supply current is flowing through the current dependent voltage source.