current analysis with pre-carried current in the inductor .- confused
When a inductor is modelled as
\$v(t) = L \frac{d\ i(t)}{dt}\$,
the reference polarity of the current is an arrow pointing from +ve part of \$v(t)\$ to -ve part.
In the question, the current reference direction is marked in the opposite direction. Hence the equation for these inductors would be
\$v(t) = - L \frac{d\ i(t)}{dt}\$.

Above image taken from slide 24 for a reference.
Note
The fact that the current values are negative and the voltage values are negative (-1800V), do not affect how the equations are setup. The numerical values are substituted after setting up the equations. The negative sign appears while setting up the equations. Th screen shots do not show the details as two separate steps (writing equations in terms of symbols and then substituting values).