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Diff Eq problem

  1. Oct 7, 2004 #1
    Here is the problem:

    Determine the orthogonal trajectories of the given family of curves.

    [tex] y = \sqrt{2\ln{|x|}+C} [/tex]

    This is what I've done so far:

    [tex] y = (2\ln{|x|}+C)^\frac{-1}{2} [/tex]

    [tex] y' = -1/2(2\ln{|x|+C)(2/x) [/tex]

    Now I understand to find the orthogonal lines I need to divide -1 by whatever I get, the problem is, I can't simplify this derivative.

    I've messed around with it a bit, and I have this:

    [tex] -(2\ln{|x|}+C)/x [/tex]

    How else can I simplify this?

    Thanks.
     
  2. jcsd
  3. Oct 8, 2004 #2

    Tide

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    First, your derivative is incorrect. Your final result may be written as

    [tex]y' = \frac {1}{xy}[/tex]

    Does that help?
     
    Last edited: Oct 8, 2004
  4. Oct 8, 2004 #3

    ehild

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    Why is the power negative?

    There is an easier way. Just square the original equation, and differentiate with respect to y.

    [tex]y^2=2\ln{|x|}+C[/tex]

    [tex]2yy'=\frac{2}{x}[/tex].
    ....
    express y', take the negative reciprocal and you get the differential equation for the trajectories. Solve, it is easy.

    ehild
     
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