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Math Help - help with orthogonal trajectories

  1. #1
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    help with orthogonal trajectories

    I tried solving for the orthogonal trajectories of the following three families:
    1. y = mx
    2. xy = c^2
    3. x^\frac{3}{2} + y^\frac{3}{2} = a^\frac{3}{2}


    here are my solutions:
    help with orthogonal trajectories-img017.jpg
    and my answers for the respective problems were:
    1. y^2 + x^2 = 2c
    2. y^2 - x^2 = 2c
    3. \sqrt{y} - \sqrt{x} = \frac{c}{2}


    But the answers from the book are, respectively:
    1. y^2 + x^2 = a^2
    2. x^2 - y^2 = a^2
    3. \sqrt{y} - \sqrt{x} = \sqrt{a}


    What am I doing wrong or not understanding or how do I make my answers more consistent with the proper format as per in the book? Especially in the 2nd problem on my left hand side i got y^2 - x^2 as opposed to the answer's left hand side which is x^2 - y^2 please help me
    He is changing the constant 'c' into 'a' on purpose? and if so then how do you know what you're suppose to change 'c' into?
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  2. #2
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    Re: help with orthogonal trajectories

    You did all 3 fine. One constant is as good as another.
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  3. #3
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    Re: help with orthogonal trajectories

    I have a new set of roadblocks with this topic of orthogonal trajectories. Here are three problems that I tried to solve:
    help with orthogonal trajectories-img018.jpg
    The first one I was able to solve however my answer differs from the answer in the book and I cannot find a way to how that they are both equal so either I or the book have the wrong answer, the second one I am stuck in a part where the equations suddenly seem to become quadratic and I do not know how to deal with this in differential equations, the third one I simply do not know how to solve the resulting differential equation after dy/dx has been replace with its negative reciprocal. Any help would be appreciated.
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