1.5 The inverse temperature as an integrating factor

The phrase ”integrating factor” origins from the theory for solving differential equations. The factor $1∕T$ is called an integrating factor for $\delta Q$, since $dS=\delta Q∕T$ is a ”total differential”.
A simple example may illustrate this:
Let

thus

We search for solutions $F(x,y)=const.$, but only know the deviation

and after transformation

The four equations are equivalent to some extend, but we lost the factor 1/x in the last equation.
For the solution $0=2ydx+xdy$ it is hard (impossible) to find a function with $dG(x,y)=2ydx+xdy$
(try?!?). We first have to multiply with the factor $x$.
Same as in the above example only after multiplying with the integrating factor $1∕T$ a total differential is found