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Math Help - Orthogonal Trajectories and Cauchy-Riemann Equations

  1. #1
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    Orthogonal Trajectories and Cauchy-Riemann Equations

    Hello,

    This is a snippet from a book:

    Orthogonal Trajectories and Cauchy-Riemann Equations-ortho_traject_cauchy.png

    I am trying to apply the same principle to:

    x^2+(y-c)^2=1+c^2

    Rearranging:
    u(x,y)=\frac{x^2}{y}+y-\frac{1}{y}=2c

    Then:
    \frac{\partial v}{\partial y}=\frac{\partial u}{\partial x}=\frac{2x}{y}
    \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}=\frac{x^2}{y}-1-\frac{1}{y^2}

    But from another thread, a function that satisfies both partial derivatives cannot be found.

    I don't know the best way to link to a another thread but here it is anyway:


    I am wondering if using the Cauchy-Riemann equations for Orthogonal Trajectories doesn't work in general or if I am missing something?
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  2. #2
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    Ok, this is how I'm interpreting your question: If you have a complex-analytic function f(z)=u(x,y)+i v(x,y) then the families u(x,y)=c are orthogonal to the families v(x,y)=k. And if you're given a harmonic function g(x,y) then you can find it's complex conjugate h(x,y) so that the function f(z)=g(x,y)+i h(x,y) is analytic and so the families g(x,y)=c are orthogonal to the families h(x,y)=k.

    Your function f(x,y)=x^2+(y-c)^2-c^2-1=0 is not harmonic so you can't use the CR equations to find the families orthogonal to it. In your exercise, you're given e^x\cos(y) which is harmonic so that the CR equations could be used to find the orthogonal families to it. However, you can still find the orthogonal famililies to x^2+(y-c)^2-c^2-1=0 and it turns out to be an interesting problem in DEs: differentiate throughout to get:

    \frac{dy}{dx}=\frac{2 x y}{x^2-y^2-1}

    and therefore the orthogonal families satisfy:

    \frac{dy}{dx}=-\frac{x^2-y^2-1}{2xy}

    which you can solve by finding an integrating factor which I think is 1/x^2.

    . . . did it kinda' quick. Work out the bugs if any ok.
    Last edited by shawsend; March 18th 2010 at 07:21 AM. Reason: corrected de and orthogonal de
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  3. #3
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    The integrating factor of 1/x^2 is spot on

    I get the solution y^2+x^2+1=cx (c a constant).
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