What would be the voltage across the across the inductor just after time t=π/wo if the diode wasn't present?
The behavior till t=π/wo should be same with or without the diode (if the diode is ideal.) As you can see in the book, the diode does not appear in the equations. Just after this instance,the capacitor voltage would still be 2*Vs as it cannot change instantaneously. So you are right, the inductor voltage has to be -Vs just after t=π/wo. After that, in the ideal situation, the voltage and current would continue to ring indefinitely as there is no resistance to damp the circuit.