When solving a Differential Equation one often encounters Constants of integration that can be quite bothersome. from what I understand, a constant can be written as another constant as long as the mapping that connects them is a a surjective one. I tend to carry out the constants out until the end because I am not sure what type of relationship will emerge. So my question is, If I end up with a solution where the same constant appears in different places, would I have to replace them with the same new constant? for example:
$$\pm cx + c^2 = y$$
The same surjective mapping must be used for both the constants? I couldn't necessarily say:
$$\pm cx + c^2 = y =kx + k$$
I am a little fuzzy with the rules used in such problems. I have not taken any group theory or anything like that. Anyways if someone could please clarify what goes on behind the scenes I would be forever in debt. Thank you for your time.