a$.$ Find $\backslash (\backslash$mathbf$\{$I$\}\_\{\backslash$mathbf$\{$X$\}\}(\backslash$mathbf$\{$t$\})\backslash )$ for $\backslash (\backslash$mathrm$\{$t$\}<mn>0</mn><mtext></mtext><mi>\backslash </mi><mi>)</mi>$ and $\backslash (\backslash$mathrm$\{$t$\}\backslash$geq $0\backslash ).$

b$.$ Verify this circuit operation in EveryCircuit, and include a screenshot visualizing the voltage vs$.$ time across the capacitor. A few suggestions to help with this:

$-$ Click on the node you want to view versus time, and the 'Show Waveform' $($Eye icon$)$ will allow you to plot the voltage versus time once the simulation is running.

$-$ You may want to increase the 'Simulation Speed' $($Clock icon once simulation is running$)$ to view the transition more quickly.

$-$ The three$-$terminal switch $($SPDT in library$)$ can be toggled while the simulation is running.

$-$ After you pause the simulation, you can reset it use the 'Rewind' option $($Back icon$)$ to experiment with simulation settings, switch settings, etc. until you can see the desired circuit behavior.