How can I solve the equation x-esin(x)=4?

[PainterGuy]

Hi,

How can I solve the following equation analytically or symbolically? "e" is Euler's number. Instead of "4", you could use the letter "c" for a constant. Could you please help me? Thank you.

 

[mfb]

You can't.

The long answer: There is no closed form for solutions to equations like that. The same problem appears for x+e^x=c and similar. There are functions defined as "let y be the solution to this problem" (like the Lambert W function, you can express x in terms of these functions, but that is not really an answer.

In some cases there are analytic solutions that can be found (e.g. x+3sin(x)=0 has a trivial solution), but that doesn't work in general.

 

[PainterGuy]

Thank you. So, now I see why Newton and others were trying other ways to find the solution: http://articles.adsabs.harvard.edu/pdf/1882MNRAS..43...43A