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In college and later the first year of university(Engineering), I was taught that you can multiply the constant of integration by a constant value and it doesn't change, like in these examples:

$$ y = \frac{1}{5}\int dx = \frac{x+C}{5} = \frac{x}{5} + C $$

$$ y = e^{\int dx} = e^{x+C} = Ce^x $$

I get what's happening here, but is it considered bad form to do this? Should I create another constant, say $K$, and let this equal (in the first example) $\frac{C}{5}$ so I can say that $y=\frac{x}{5}+K$, or is it just accepted that it's slightly iffy but everyone understands what you've done?

Flyingfirepig
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    It's misleading and dangerous, because you really shouldn't use the same letter to mean two different things within the same line. However, people do it all the time, so c'est la vie. –  May 07 '18 at 16:04
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    The only thing that is really any significant danger is losing track of changes in constraints on constants. For example, strictly speaking in your case, the first and second $C$s are arbitrary but the third $C$ is positive. This can be confusing because in fact the third $C$ is arbitrary but the calculation you did does not show that. Other than that, the only possibility is if there is some clear interpretation of one of the constants of integration (e.g. $C=y_0$), in which case you should probably carry it through in terms of its interpretation. – Ian May 07 '18 at 16:16
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    This is closely related to another stylistic point, which is that you often want to avoid constants of integration altogether, opting to instead explicitly solve for them in terms of givens of the problem. – Ian May 07 '18 at 16:19
  • @Ian I have heard people who don't believe in the existence of indefinite integrals. I must admit that the view has its merits. – Arthur May 07 '18 at 16:44
  • @Ian: actually, in some cases, keeping track of those constraints makes you lose information. Typical example would be when solving $y'-y=0$, the solution is $e^{t+c}$, so one would write $de^t, with $d>0$. But it turns out that $d\leq0$ also produces valid solutions. One of several problems of trying to avoid the use of complex numbers. – Martin Argerami May 07 '18 at 18:38
  • @MartinArgerami I pointed this out. The problem is that we write things under one set of assumptions and then write conclusions that require entirely different assumptions. For example, when solving $y'=y$ by separating variables, even if you include absolute values in logarithms, the solution $y=0$ is still not actually found by the separation of variables procedure. – Ian May 07 '18 at 18:42
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    That's called a "generic constant" and actually used by professional mathematicians from time to time. Usually, you announce, at the beginning of proof "In this proof $C$ will denote a generic constant that can change its value from line to line." – Dirk May 07 '18 at 18:49

3 Answers3

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It is considered bad form. The usual approach, at least in a final presentation (as opposed to when you're actually doing the calculations), is to know how many different $C$s you need, and use either $C',C''$, etc. until you get to the final one, which is just $C$ (if there aren't too many, $C'''''$ is a bit ridiculous), or use indices: $C_1,C_2,$ etc.

If what you're writing is only meant for your eyes, you can really do whatever you feel like; mathematical notation conventions are there to facilitate communication between people, not to put restrictions on what people write down as their own personal notes. That being said, eliminating a potential source of confusion at nearly no cost of writing speed or cognitive load sounds like a good thing to me, so I would suggest you use something like this in those cases as well.

Arthur
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    A rule of thumb for even one’s own private notes is “am I going to understand this six months from now?” – amd May 07 '18 at 16:59
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    However, there are certain fields of mathematics (like PDE) where there would be so many different constants that the standard is to use $C$ as a single “generic” constant whose value changes all the time. – Hans Lundmark May 07 '18 at 17:59
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    @HansLundmark True. In that case, $C$ doesn't directly represent a constant, just a placeholder saying "some constant goes here". – Arthur May 07 '18 at 18:02
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    Alternatively, we can take $C$ to mean the whole (one-dimensional) vector space of constants, and the integral as a member of the quotient space of all functions (or classes of functions) mod $C$. But I don't think I've ever heard or seen anyone say that explicitly. – tomasz May 07 '18 at 23:08
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As Arthur's answer has pointed out, in general you should clarify that your variable is changing. However, in this particular situation (solutions to differential equations) it is very commonly understood that C isn't meant to be understood as a normal variable, but rather as a placeholder to mean "some constant" (often called a generic constant). This is of course a fundamental departure from normal equations, but is so useful in situations where you would need way too many different C's that it is often accepted and understood. As ever with notation, consider who your audience will be and what they are used to, and clarify when necessary.

Heisenbugs
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Arthur's answer is excellent, but I think you'd find it useful to see a situation with a previous question on math.stackexchange where re-using the same variable to represent constants lead to much more subtle problem. The question is here.

Even though they (correctly) used $C_1$ and $C_2$ to represent arbitrary constants when solving a diff-eq, they re-used those same variables for another function that was related to the first function through some equations, and ended up with an incorrect final answer. (My answer giving a full description of what went wrong is here. )

JonathanZ
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