When shown equation $(1)$, I have two different answers for its integration, one mine, one more from a colleague and I am uncertain of which is the correct one.
$$\left( \frac{\partial r}{\partial T}\right)_{E/T}- r\frac{c_0}{T}= - \frac{c_0}{T} \tag{1}$$
where the subscript indicates a constant ratio of $E/T$ throughout the calculations.
My take on it:
Using an integrating factor $I= e^{\int P dT}$ where $P=Q= \frac{c_0}{T}$.
Following the rule for this method:
$$I r = \int^{T_f}_{T_0} I Q dT $$
$$\left( \frac{T_f}{T_0}\right)^{-c_0}r= \int^{T_f}_{T_0} \left( \frac{T_f}{T_0}\right)^{-c_0} \left( \frac{-c_0}{T} \right) dT + f(E/T)$$
because $E/T$ is seen as a constant and would be differentiated to $0$ I added a function of this term in my calculation, $f(E/T)$.
$$r=\ln \left( \frac{T_f}{T_0}\right)^{-c_0}+ \left(\frac{T_0}{T_f}\right)^{-c_0} f(E/T)$$
My colleague's take on it:
I don't understand where his answer comes from, but he said to have used the same process of integration, using an integrating factor, and choosing $K$ as the constant term.
$$r = -e^{\int_{T_0}^{T} c_0/T^\prime dT^\prime} \int \frac{c_0}{T^\prime} e^{-\int c_0/T^{\prime \prime} dT^{\prime \prime}} dT^\prime - K e^{\int_{T_0}^{T} c_0/T^\prime dT^\prime}$$
Which is the correct integration using an integrating factor?
The flaw is in your method, not (necessarily) your colleague's (although this really depends on what the bounds are for his antiderivatives, it's really hard to tell from the clutter in his notation). You treat the integrating factor like a constant, when it really should have been $$I = \left(\frac{T}{T_0}\right)^{-c_0} \neq \left(\frac{T_f}{T_0}\right)^{-c_0}$$ As a sanity check, nontrivial integrating factors should always be varying functions, not constants (otherwise, what's the point of an integrating factor?)