Can x be found for ax^b - x^(b-2) = c in the general form?

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Discussion Overview

The discussion centers around the equation axb - x(b-2) = c, focusing on the possibility of finding a general solution for x. Participants explore various conditions and methods related to solving this equation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses difficulty in solving for x after simplifying the equation.
  • Another participant suggests that a closed form solution may not exist, proposing the use of infinite fractions or similar methods as potential approaches.
  • It is noted that a solution for x might be possible only if b is less than or equal to 4.
  • Further conditions are mentioned, indicating that b should also be greater than or equal to -2 and an integer, while also noting that special cases such as c=0 or a=0 do not assist in solving the general problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a general solution for x, with multiple competing views and conditions presented regarding the parameters of the equation.

Contextual Notes

Participants highlight specific conditions under which solutions may or may not exist, but these conditions remain unresolved in terms of their implications for the general case.

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I have an long equation I managed to simplify to that form, but now I haven't the foggiest on how to solve for x, any help appreciated
 
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Not in a closed form. There might be some way with infinite fractions or similar approaches.
 
Only if ##b\le 4##.
 
... and ##b \geq -2## and an integer.

Or c=0 or a=0, but none of these special cases help with the general problem.
 

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