This question may be overly simple, but why does the integration constant appear in some versions for the catenary equation, while in others, not so. What is the mathematical significance of the integration constant, i.e. what does the $C$ in $y = a\cosh(x/a) + C$ denote?
In a nutshell, $C$ is not so much the constant of integration, but more or less determines the depth or placement of the catenary.
If $C>0$, the catenary extends further up from the origin up the $y$-axis, while if $C<0$, the catenary extends further down from the origin down the $y$-axis. If $C=0$, the inflection point of the catenary will be at $(a,0), a \neq 0$.