Can you solve for y in sin(y) - y = x√(2)?

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SUMMARY

The equation sin(y) - y = x√(2) does not have a straightforward closed-form solution for y. Instead, it can be approached using the Lambert W function, which is a key tool in solving transcendental equations. The discussion highlights the relevance of this equation in celestial mechanics, particularly in relation to the Kepler equation. For practical solutions, iterative methods, power series, Fourier series, or numerical interpolation are recommended due to the periodic nature of the function.

PREREQUISITES
  • Understanding of transcendental equations
  • Familiarity with the Lambert W function
  • Knowledge of iterative methods for numerical solutions
  • Basic concepts of celestial mechanics and the Kepler equation
NEXT STEPS
  • Research the properties and applications of the Lambert W function
  • Learn about iterative methods for solving transcendental equations
  • Explore power series and Fourier series techniques for function approximation
  • Investigate the relationship between the Kepler equation and periodic functions
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Mathematicians, physicists, and engineers interested in solving transcendental equations, particularly in the context of celestial mechanics and numerical analysis.

wheepep
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sin(y) - y = x√(2)
solve for y
 
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What makes you think a closed-form solution for y can be found?
 
Maybe there exists a closed form solution exists, maybe not... But could I ask, what is the purpose for searching such solution?

I know, in celestial mechanics, the Kepler equation is same as this - at least after a change of variables - which means, you can take a look at literature, if you find something about it. As being so, I wouldn't use my time to kick the equation, as it is one of the most researched one in the world. Unless I wanted some sort of challenge, of course.

Anyhow, your choices for the solution will most likely be an iterative method, a power series or a Fourier series or interpolation of the numerical solution over the values of x under interest.
 
In general, periodic functions are of interest due to their frequent occurrence in natural phenomenon. As speculation, this particular function may be of interest due the times and places it occurs.
 

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