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For example, a differential equation of the form $$\frac{dy}{dx}+P(x)y=f(x),$$ has the general solution $$y=\mu^{-1}\int\mu f(x)\,dx +C\mu^{-1},$$ where $\mu$ is the integrating factor $\mu(x)=\exp{(\int P(x)\,dx)}$.

So the way I've written this, the parameter for the integrating factor is suppressed when I write it down in the solution but when I defined it, I explicitly say it is a function of $x$. However, for $f(x)$, I write the parameter explicitly in the solution because it felt wrong otherwise. Further, for $y,$ I also suppressed the parameter.

My question, then, is whether or not this is the best way to write it. Is there a general consensus on when to explicitly state the independent variable in the notation? And is it a problem that I'm somewhat inconsistent on when I did it?

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"whether or not this is the best way to write it"

This depends on the audience for your (attempt) at mathematical communication. Attempts to communicate can be very successful, moderately successful, et al. It can be a heap of trouble if the audience thinks they've understood, but in fact have misunderstood.

Sometimes the audience is yourself, as in your personal notes. In my notes I write $\int f$ rather than $\int f(x) dx$ if there can be no misunderstanding among the audience members (me).

Your audience might be undergraduates, graduate students, your advisor, a roomful of people of very different backgrounds.

Opinions on "the best way to write it" will vary with the feelings of various people, to use your word.

A very general consensus is unlikely, I'd say, for the question of "when to explicitly state the independent variable".

Lastly you ask about consistency within your own presentation. I don't know if it is a major problem. I think you are following a common practice: to be more explicit when introducing a dependent quantity for the first time, but then announce that the dependence will be implicit later on. It is a trade-off. You are buying a sleeker presentation at the cost of possible misunderstanding later on, among certain members of the audience.

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