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

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The equation sin(y) - y = x√(2) raises questions about the existence of a closed-form solution for y. Some suggest that replacing sin(y) with its exponential form may lead to a solution involving the Lambert W function. The discussion references the Kepler equation, indicating that this problem is well-studied in celestial mechanics. For practical purposes, iterative methods, power series, or numerical interpolation are recommended for finding solutions. The periodic nature of the function suggests its relevance in various natural phenomena.
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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?
 
Replacing [math]sin(y)[/math] by [math]\frac{e^{iy}- e^{-iy}}{2i}[/math] you might be able to get a solution in terms of the "Lambert W function" https://en.wikipedia.org/wiki/Lambert_W_function.
 
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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