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March 9th, 2015, 03:22 PM   #1
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Show there exists a unique solution

y' = 1/(2y*sqrt(1-x^2))

Without solving the initial-value problem, argue that there exists a unique solution.

I can solve it and all, but the existence and uniqueness kind of confuse me. I think it's along the lines of

F(x,y) = 1/(2y*sqrt(1-x^2))
y=! 0, x =! +-1

Fy(x,y) = -1/(2y^2 * sqrt(1 - x^2))
y =! 0 x =! +-1

Does this just show existence for a solution? Or does it also show uniqueness? If not, what shows uniqueness?
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March 10th, 2015, 01:45 AM   #2
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For any equation of the form y' = 1/(y*f(x)), one has d/dx(y²) = 1/f(x), so a solution can be found by integrating to get y² = ∫1/f(x) dx and then taking the square root of each side, but that reasoning effectively obtains the general solution, and so is within a whisker of solving the equation for specific initial conditions. As the question specifically prohibits solving the initial value problem, I'm not sure whether it would be acceptable to proceed in that way, but I can't see an alternative.
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