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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?

  2. jcsd
  3. Oct 8, 2004 #2


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    Homework Helper

    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


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    Homework Helper

    Why is the power negative?

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


    express y', take the negative reciprocal and you get the differential equation for the trajectories. Solve, it is easy.

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