Problem $18$:

The circuit shown below has been operating for a long time with the switch in position $\backslash (\backslash$mathbf$\{$A$\}\backslash ).$ At $\backslash (\backslash$mathrm$\{$t$\}=0\backslash ),$ the switch is flipped into position B$.$ Find the time$-$dependent expression for the voltage across the capacitor, $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$C$\}\}(\backslash$mathrm$\{$t$\})\backslash ).$