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Math Help - Ode

  1. #1
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    Ode

    Has this old saw ever been answered?
    Does the differential equation:
    dy/dx=1/x+1/y
    have a solution in terms of elementary functions?
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  2. #2
    Super Member Rebesques's Avatar
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    I don't think so



    This doesn't mean we can't solve it - even approximate the solution numerically.

    As I sucked in Lebesgue spaces, I assume y is smooth, and for the sake of well-posedness, y(1)=a.

    Then y'(1)=1+\frac{1}{a}, \ y''(1)=-1-\frac{1}{a^2}-\frac{1}{a^3} etc,

    and a Taylor's expansion sais y(x)=a+(x-1)y'(1)+\frac{(x-1)^2}{2}y''(1)+...

    so

    y(x)=a+(x-1)(1+\frac{1}{a})+...
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  3. #3
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    Ode

    Thanks for responding. I appreciate your interest in this question.

    Of course you can solve it numerically! But can the numerical solution be
    shown to be an ELEMENTARY function? That's the question.
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  4. #4
    Super Member Rebesques's Avatar
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    ...Well I already said i don't think so!
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  5. #5
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    Ode

    Yes you did! And thanks again!
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  6. #6
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    I did try to solve this but alas I am having little success.

    Rewrite,
    y'-y^{-1}=x^{-1}
    Note, this is a non-homogenous.
    Thus, begin by finding the general solution of,
    y'-y^{-1}=0
    Thus,
    y'=y^{-1}
    Note, this is a Bernoulli, equation
    Use substitution,
    u=y^2
    Thus,
    \frac{du}{dx}=\frac{du}{dy}\cdot \frac{dy}{dx}.
    Thus,
    \frac{du}{dx}=2y(y^{-1})=2
    Thus,
    u=2x+C
    Thus,
    y^2=2x+C
    Thus,
    y=\sqrt{2x+C}---> general solution.

    For the particiluar solution for this ode I had trouble. I went threw many different attempt non of them work. So maybe this is no elementary function which satisfies this particular condition.
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  7. #7
    Super Member Rebesques's Avatar
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    non-homogeneous [...] particiluar solution

    I don't think that method applies here, because the equation is not linear.
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  8. #8
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    Quote Originally Posted by Rebesques
    I don't think that method applies here, because the equation is not linear.
    That bothered me too, I chose to ignore it.
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  9. #9
    Super Member Rebesques's Avatar
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    That bothered me too, I chose to ignore it.


    ...U are the man!!!
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  10. #10
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker
    That bothered me too, I chose to ignore it.
    We'll make a Physicist out of this guy yet!

    -Dan
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  11. #11
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    I only do that when it comes to diffrencial equations.
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