(Edited to add: I may have solved my problem now. See end of post.)

This does not work (edited to fix a mistake of mine):

DSolve[{
  0 == -voltageC0[t] - 2*voltageC0''[t] + 2*voltageC1''[t],
  0 == -voltageC1[t] - 2*voltageC1''[t] + 2*voltageC0''[t]
  }, voltageC0[t], t]

Mathematica replies:

DSolve::overdet: There are fewer dependent variables than equations, so the system is overdetermined.

I cannot make sense of that response from Mathematica. I see two dependent variables, and I see two equations. Two equals two. What is the meaning of Mathematica's response to my query?

Edited to add: This attempt apparently succeeds:

DSolve[{
  0 == -voltageC0[t] - 2*voltageC0''[t] + 2*voltageC1''[t],
  0 == -voltageC1[t] - 2*voltageC1''[t] + 2*voltageC0''[t]
  }, {voltageC0, voltageC1}, t]

The large constant-numerical-values appearing in the solution do surprise me though.