My Math Forum Show there exists a unique solution

 Differential Equations Ordinary and Partial Differential Equations Math Forum

 March 9th, 2015, 03:22 PM #1 Newbie   Joined: Mar 2015 From: Australia Posts: 7 Thanks: 0 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?
 March 10th, 2015, 01:45 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,036 Thanks: 2274 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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