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Sometimes in math, a 'dummy solution' like some dummy variable is used in place of some equation needing to be solved, for example. The equation may be solved for this 'dummy solution', and then that solution can be used to ascertain the general solution. I saw my professor use a term for this in a dynamical systems course but I forgot the name.

Nij
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Mike
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    Can you give a specific example? I'm not familiar with the concept. – dmh Sep 30 '22 at 05:43
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    This feels like 'without loss of generality' but I'm not sure. – soupless Sep 30 '22 at 05:47
  • substitution maybe? Not sure though. – whoisit Sep 30 '22 at 06:19
  • Like solving $X^2+X+1=0$ first instead of $x^4+x^2+1=0$ by letting $X=x^2$ ? – nicomezi Sep 30 '22 at 06:40
  • That might be "a particular solution", which consists (very loosely) of the solution for a homogeneous set of equations which can then be used to find the general solution to the full set. Used in particular for differential equations. – lmsteffan Sep 30 '22 at 06:41
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    There are too many areas which use "Similar Concepts" with varying names. In addition to what is listed in the comments, I think we can also consider (1) Non-linear Equations can be solved with Iterative methods with a "guess Solution" which will converge to a "Better Solution" (2) Maxima & Minima with constraints can be solved using Lagrange Multiplier https://en.wikipedia.org/wiki/Lagrange_multiplier – Prem Sep 30 '22 at 07:53
  • @lms, also for difference equations (aka recurrence relations). – Gerry Myerson Oct 02 '22 at 00:58

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I found the term in old notes. It is 'ansatz', which wolfram describes as "an assumed form for a mathematical statement that is not based on any underlying theory or principle. An example from physics is the 'Bethe Ansatz'".

My previous understanding of it appears to be inaccurate. I thought it was like the 'product solution' in the 'Separation of Variables' Homogenous PDE method where some variable solution is proposed and then later solved for, but an 'ansatz' appears to be not this at all.

The website calculushowto.com says "These equations can’t be solved by a single method, so you could use a guess and check method. If your guess and check results in a solution for the differential equation, then you’ve just used an ansatz. Note though, that your initial guess can’t be based on a known theory or principle. So if you’re following, for example, the separation of variables method, that isn’t an ansatz."

Mike
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