How do I solve the Brachistochrone Problem with a given differential equation?

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SUMMARY

The discussion focuses on solving the differential equation (1 + y')²y = k², which is a key component of the Brachistochrone Problem. The user, 930R93, expresses difficulty in finding a productive method for integration. A suggested approach includes integrating y' = √((k² - y)/y) and utilizing the substitution y = k²sin²(θ) to simplify the problem. This method provides a clear pathway to tackle the Brachistochrone Problem effectively.

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930R93
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Hello,
I'm having problems with a D.E. question,
I'm asked to solve the equation:
[tex]\left(1+y^{'2}\right)y=k^{2}[/tex]
where K is a certain positive integer to be determined later.
This more commonly known, as you probably know, as one of the solutions to the Brachistochrone Problem.
I really have no idea where to start. I've experimented with various methods of integration, which create nothing which I recognized as productive.
A gentle prod in the right direction would be much appreciated!
Thanks!
-930R93
 
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Integrate [tex]y'=\sqrt{(k^{2}-y)/y}[/tex]. It is useful to use [tex]y=k^{2}sin^{2}(theta)[/tex] .
 

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