Consider a RC circuit functioning as a high pass filter. The transfer function is given by:
For a frequency of 0, the transfer function is 0. Which means for DC input signals, the output response is 0.
However, by modelling the circuit with a differential equation and then solving it shows that the voltage across the resistor in a RC circuit is actually a decaying exponential.
Isn't this a contradiction? The transfer function shows that the output should be 0, but the differential equation shows that it's a decaying exponential?