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This differential equation came up in a problem I'm working on. I've never seen one with the function or its inverse in a limit of integration. Can anyone point me in the right direction (or maybe even show me how to solve it)? Thanks.

$y^{\prime }(x)[1-x]^{2}x-\int_{y^{-1}(x)}^{x}(y(z)-x)2zdz=0$

Initial condition: $y(1) = 1$

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If you replace $x$ by $y(x)$ you get rid of the inverse function. But do you have a reason to believe this DE should have solutions? It's a type of delay DE but it's a variety of delay DE for which I'm not sure whether or not you can expect solutions. – Ryan Budney Nov 5 '10 at 16:52
In fact, there may not be a solution (I apologize, I should have mentioned that). If not, any hint about proving that it has no solutions would be valuable as well. NOTE: I added an initial condition that I omitted in the original post. Thanks Ryan. – Brett Katzman Nov 5 '10 at 17:02
Just a guess, but supposing some nice solution exists, could you try a power series solution? So long as y(0)=0 (no constant coefficient), the power series should be invertible. Then it looks like you have to play the coefficient matching game. – Alex R. Nov 6 '10 at 20:36

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