Laplace Transform using Impedances and Initial conditions

Full credit will be given for a detailed, well explained and neatly written solution only

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The input to the circuit shown in Figure below is the voltage of the voltage source, $18$ V $.$ The output of this circuit, the voltage across the capacitor, is given by

$v_0(t)=6+12e^{-2t},V$ when $t>0$

Figure

Determine $Vo(0)$ from the circuit

Calculate $Vo()$ from the given equation for $Vo(t)$

Draw equivalent circuit for $Vo()$ and write an equation for $Vo()$

Find R value in Ohms comparing steps $2$ and $3.$

Draw circuit diagram in s$-$domain for $t>0$ including impedances and initial conditions

Label circuit in s$-$domain

Analyze the circuit in s$-$domain to find equation for the voltage Vo$($s$)$ across the capacitor. Show your work in detail

Take the reverse Laplace transform to obtain equation for $Vo(t)$ across the capacitor in time domain

Find the capacitance comparing $Vo(t)$ to the given equation

Determine time constant in seconds

Compute $Vo(t)$ at $t=1s,$

Find equation for the current through the capacitor and compute its value at $t=1s$