Consider the following special case with T(t) = T_0 for t < 1970, T(t) increasing linearly for 1970 < t < 1998 to the value T_1 with T(t) = T_1 for t > 1998. The corresponding solution C(t) of the equation dC/dt increases linearly for t < 1970, quadratically for 1970 < t < 1998 and again linearly for t > 1998 as sketched by the solid lines in the following graph:
We see that after 1998 the temperature stays constant while the CO2 increases linearly. The solid lines could picture reality.
On the other hand, if you want to create a fiction of CO2 alarmism, you would argue as follows: Look at the solid lines before 1998 representing recorded reality and simply make an extrapolation until 2020 of the simultaneous increase of both T and C during the period 1970 - 1998, to get the dotted red line as a predicted alarming global warming in 2020 resulting from a continued increase of CO2. The extrapolation would then correspond to using a connection between T and C of the form T ~ C with T determined by C, instead of the as in the above model dC/dt = T with C determined by T.
This shows the entirely different global warming scenarios obtained using the model T ~ C with T determined by C, and the model dC/dt = T with C determined by T.