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Math Help - C^1 solutions intersecting circles

  1. #1
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    C^1 solutions intersecting circles

    Given the following C^1 solution (x(t),y(t)) of:
    {x' = -x^3 +x*y^2
    {y' = -y^3 +y*x^2

    How do I show that it will not interesct a circle centered @ (0, 0) more than once?

    Also, I cannot find C^1 solution in my book. What is it?
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Borkborkmath View Post
    Given the following C^1 solution (x(t),y(t)) of:
    {x' = -x^3 +x*y^2
    {y' = -y^3 +y*x^2

    How do I show that it will not interesct a circle centered @ (0, 0) more than once?

    Also, I cannot find C^1 solution in my book. What is it?
    C^1 means that the function is differentiable once and its derivative is continuous.

    Note that

    \displaystyle \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}=  \frac{-y^3 +yx^2}{-x^3 +xy^2}=\frac{-y(y^2-x^2)}{x(y^2-x^2)}=-\frac{y}{x}

    This can be solved by separation of variables

    \displaystyle \frac{dy}{y}=-\frac{dx}{x} \iff \ln|y|=-\ln(x)+\ln(c) =\ln\left( \frac{c}{x}\right) \implies y=\frac{c}{x}

    So unless there is a typo a hyperbola can intersect a circle centered at the origin more than once.
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  3. #3
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    When I put
    {
    {

    I meant that as a piecewise function, if that changes anything. I think that the professor might have typo'd the question if, as you beautifully explained it, the equation can be separated by variables.

    On a side note, how do you type with all the mathematical symbols?
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  4. #4
    Behold, the power of SARDINES!
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    2nd question first. It is called LaTex
    http://www.mathhelpforum.com/math-help/f47/

    I assume you mean parametric form. I just eliminated the parameter. If the minus sign was not there you would get a line though the origin. If there were some restriction like
    x > 0 or y > 0then the statement would be true.
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