Charge on plates of two capacitors in series are not equal
One can make any number of separated conductive objects (plates, and
in the case of your diagram, the plate pair of C2(negative) and C1(positive)
is a single conductive object).   One can then apply a voltage difference
with a battery, and remove the battery (open the switch).

The C2(negative) charge PLUS the C1(positive) charge, if it started at zero,
would still be zero after the battery connection (which doesn't add charge to
the plate pair).   But while the sum of those charges might be zero,
that is not in fact expressed in any of your formulae.   Did you
wish to add that as an initial condition?