** I'm a student.
First I'd like to show
why we construct the s-domain circuit like that in the second diagram in post #6:
View attachment 95650
We can treat the first constant term as a voltage source, the second the voltage drop across the capacitor, the third across the resistor and the fourth across the inductor, so the equation itself indicates a kind of Kirchhoff's voltage law, and we build the s-domain circuit from it.
Next we can solve for I from the last equation in terms of s and the capacitor's initial voltage, then Vo is simply equal to IsL.
Once we have Vo in the s-domain, we can change it back to the corresponding time domain solution by reverse Laplace transform.
Hope it helps.