How Do You Solve This Equation?

Discussion in 'Physics & Math' started by Eugene Shubert, Mar 18, 2010.

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  1. Eugene Shubert Valued Senior Member

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    Find the most general non-trivial differentiable function f that satisfies

    f(x)f(y) = f(x*sqrt(1-y^2) + y*sqrt(1-x^2))

    for all x and y in the interval [-1, 1].
     
    Last edited: Mar 18, 2010
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  3. temur man of no words Registered Senior Member

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    Try e.g. x=cos(u), y=cos(v).
     
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  5. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    Normally I wouldn't respond to posts like this, but I can get things into the form

    \(\partial_x z(x,y) \partial_x g(x,y) = \partial_y z(x,y) \partial_y g(x,y)\)

    assuming you can interchange a few derivatives, where

    \(z(x,y) \equiv x\sqrt{1-y^2}+y\sqrt{1-x^2}\)

    and

    \( g(x,y) \equiv \partial_z f(z)\).

    Then you can try separation of variables \(g(x,y) = \xi(x) \psi(y)\), and plug it into the differential equation, and see if you can get one side as only a function of x and the other side as only a function of y, something like

    \(F(x) = G(y)\)

    where F and G will involve derivatives of functions, but will only be functions of x or y. This might be a bit tricky, because z is not a ``nice'' function.

    After all fo this, I see temur has responded:

    This is probably much easier

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  7. Eugene Shubert Valued Senior Member

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  8. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    NOTE TO ALL POSTERS:

    We're here to help, not to do your homework for you.

    You did three lines of algebra since temur posted his hint, and now you're asking for help again?

    Think for yourself.

    Thread closed.

    PM me if you want it reopened, and we can talk.
     
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