Solving x²=-1/2ln(x) with x in (0,2]

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Discussion Overview

The discussion revolves around solving the equation x² = -1/2 ln(x) for x in the interval (0, 2]. Participants explore methods for finding solutions, including numerical approaches and special functions.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant expresses difficulty in solving the equation x² = -1/2 ln(x) within the specified interval.
  • Another participant suggests that equations of this type cannot be solved exactly using elementary functions and recommends numerical methods like Newton's Method for approximation.
  • A third participant references the Lambert W function as a potential tool for solving the equation.
  • Another participant proposes the Wright omega function as an alternative approach suitable for the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific solution method, with multiple competing approaches being suggested, including numerical methods and special functions.

Contextual Notes

The discussion does not resolve the mathematical steps involved in applying the suggested methods or the implications of using different functions.

Paragon
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Hi,:-p

so, I can't solve the embarissing:

[tex]x^{2} = -\frac{1}{2}ln(x)[/tex]

, where [tex]x \in ]0, 2][/tex] or [tex](0 < x \geq 2 )[/tex]

any hep would be nice...
thanx for your pacience!
 
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Paragon: Equations of this kind can't be solved exactly, i.e the solutions can't be expressed in elementary functions. You can use a numerical method, such as Newton's Method, to approximate the solution.
 
Hi, why don't you try with Wright omega function it suits your requirement

y+lny=z
or as suggested by others use LambertW function
 

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