Suppose I want to graph the general solution to a real first-order ODE as a surface in $\mathbb{R}^3$. What are the correct axes of the space in which this surface lives?
My first guess was that the horizontal axes represent the independent and dependent variables, and the vertical axis represents the constant of integration, but this has bizarre effect of "de-arbitrizing" that arbitrary constant. For example, the general solutions $x^2 + y^2 = C$, $x^2 + y^2 = -C$ and $x^2 + y^2 + 500 = -e^{\pi + C}$ all have different graphs under this graphing methodology, even though they are the same general solution.