I can't say where you went wrong with your math but, in this simple example of an RC network, whenever the voltage source rises to 5 volts from 0 volts or falls to 0 volts from 5 volts, that "edge" passes through the capacitor as if the capacitor is 0 Ω or a short circuit.
Hence, the first time the 5 volts drops to 0 volts, the voltage that the output has decayed to then drops instantly by 5 volts. So, where you have calculated the voltage having dropped to 1.84 volts (during the first 5 volt presence of the input), the instant that voltage drops to 0 volts, the +1.84 volts must also drop instantly by 5 volts to -3.16 volts.
You have it dropping from +1.84 volts to -1.84 volts and clearly that cannot be correct because it should be a 5 volt step change.