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I have heard the words "trivial" and "non-trivial" a lot in reference to all sorts of mathematics. To give you an idea of what I mean, consider the following differential equation:

$$(y'')^x-x^2y'+y^2=0$$

This differential equation most likely has no "non-trivial" solution in terms of elementary functions, but one can easily guess the solution $y=0$, which is called a "trivial" solution.

My question is,

Is there a rigorous mathematical definition for "trivial" and "non-trivial"?

Tdonut
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    No. Why would we expect there to be? – Thomas Andrews Jun 08 '15 at 15:18
  • @Thomas I thought there might be one because I have heard it a lot; its useful to define things before discussing them, in my opinion. – Tdonut Jun 08 '15 at 15:20
  • It all depends on context. In the case of differential equations, anything constant might be considered "trivial." But there are lots of other cases. Fermat's Last Theorem is often expressed in terms of "non-trivial" solutions to $a^n+b^n=c^n$, for example. It's just. Loose way of saying, "Yeah there is an obvious solution/set of solutions, but..." – Thomas Andrews Jun 08 '15 at 15:22
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    The term is highly overloaded. Some denotations can be made rigorous (e.g. trivial divisor), some cannot (e.g. "the proof is trivial"). – Bill Dubuque Jun 08 '15 at 15:24
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    The word trivial is used often in mathematics. Sometimes it is rigorously defined, as in the trivial topology, or trivial subgroup. These definitions are, of course, highly dependent on the context. When the word trivial is used for proofs, it is completely subjective however. What is trivial to one person may not be to another. – Tim Raczkowski Jun 08 '15 at 15:25
  • This question may have been sparked by the recent question What exactly is a trivial module? – Bill Dubuque Jun 08 '15 at 15:36
  • The definition of 'trivial' is trivial, hence we assume it and further proceed to prove... – shardulc Jun 08 '15 at 16:53
  • @Bill That is an interesting question, but this question was not "sparked" by that one. – Tdonut Jun 08 '15 at 16:53

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