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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.

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up vote 3 down vote accepted

I realize it is a nebulous (yet precise!) answer, but here goes: just do what is mathematically correct.

For example, in the instance you gave, going from $$y=\pm cx + c^2$$ to $$y=kx+k$$ doesn't hold generally since you are replacing the quantity $\pm c$ and the quantity $c^2$ both by the same constant $k$, when it's not necessarily the case that $\pm c=c^2$.

PS If you could provide another example or two where you are unsure of what constant(s) you can/cannot consolidate, we can probably shed more light on the issue for you through those examples.

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No I think helped me out a great deal, thank you very much – Cactus BAMF Feb 16 '13 at 19:14
@CactusBAMF: Let me also add that while it would not be wrong to replace $y=\pm cx+c^2$ with $y=Ax+B$, you would have "lost" some information by doing so, e.g. the particular relationship between $A$ and $B$ here and the fact that $B\ge 0$. – JohnD Feb 16 '13 at 19:22

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