The effect of an inductor is to keep the current at the same value. When the battery is disconnected after reaching the steady state, inductors $L_1$ and $L_2$ become sources of current. The voltage across each is such as to keep the same current flowing through each while ensuring that Kirchhoff's Rules are satisfied.
So the way to tackle this problem is : (1) find the currents through each inductor when the battery is connected; (2) after the battery is disconnected, imagine each inductor to be a battery of such voltage that the current through each is the same as before. The sum of the voltage drops around each of the 2 loops that remain must be zero. This enables you to find the PD across each inductor.
I agree with the answers given $(L_1=\frac32V, L_2=V)$.
It may seem wrong that voltages in the circuit can be greater than the applied voltage $V$ - or even equal to $V$ when there are resistors in the circuit. In an RC network the voltage on a capacitor could never exceed that of the source. However, large voltages (back emfs) can be generated by inductors when the applied current is suddenly changed.