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

  • MHB
  • Thread starter wheepep
  • Start date
In summary, the conversation discusses the possibility of finding a closed-form solution for the equation sin(y) - y = x√(2). It is mentioned that a solution may be found using the Lambert W function, but it is uncertain if such a solution exists. The purpose of finding a closed-form solution is questioned, and it is suggested that the equation is similar to the Kepler equation in celestial mechanics. The use of iterative methods, power series, and Fourier series are also mentioned as possible solutions. It is speculated that the function may be of interest due to its periodic nature and its occurrence in natural phenomena.
  • #1
wheepep
9
0
sin(y) - y = x√(2)
solve for y
 
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  • #2
What makes you think a closed-form solution for y can be found?
 
  • #3
Replacing \(\displaystyle sin(y)\) by \(\displaystyle \frac{e^{iy}- e^{-iy}}{2i}\) you might be able to get a solution in terms of the "Lambert W function" https://en.wikipedia.org/wiki/Lambert_W_function.
 
  • #4
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.
 
  • #5
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.
 

1. What is the purpose of solving for y in this equation?

The purpose of solving for y in this equation is to find the value or values of y that satisfy the equation. This can help in understanding the relationship between the variables x and y, and can also be useful in solving other equations involving trigonometric functions.

2. Is it possible to solve for y in this equation?

Yes, it is possible to solve for y in this equation. However, the solution may not always be possible to express in a simple form, and may require the use of numerical methods or approximations.

3. What are the steps to solve for y in this equation?

The general steps to solve for y in this equation are as follows:

  1. Isolate the term with the trigonometric function (sin(y) in this case) on one side of the equation.
  2. Use inverse trigonometric functions to eliminate the trigonometric function and solve for y.
  3. If there are multiple solutions, consider the domain and range of the equation to determine the appropriate solutions.

4. Are there any special cases to consider when solving for y in this equation?

Yes, there are a few special cases to consider when solving for y in this equation:

  • If the equation has multiple trigonometric functions, it may be helpful to use trigonometric identities to simplify the equation before solving for y.
  • If the equation has a square root, it is important to consider both the positive and negative roots when solving for y.
  • If the equation has a coefficient in front of the trigonometric function, it may need to be factored out before solving for y.

5. Can this equation be solved algebraically?

Yes, this equation can be solved algebraically. However, as mentioned earlier, the solution may not always be able to be expressed in a simple form and may require the use of numerical methods or approximations.

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