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Orthogonal trajectories

  1. Sep 16, 2009 #1
    The set of orthogonal trajectories for the family indicated by
    ( x-c)^2 + y^2 = c^2

    My work:

    y' = -(x-c)/y Since c= ( x^2 + y^2 ) / 2x

    plugging back in and doing -1/y' i got

    y' = 2xy / ( x^2 - y^2)

    Then im supposed to move the x and y to a side and integrate but i don't see how it is possible.
  2. jcsd
  3. Sep 16, 2009 #2


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    Science Advisor

    You can't. This is NOT a "separable" differential equation. It is, however, a "homogeneous" equation because replacing x by a x and y by ay on the right gives 2(ax)(ay)((ax)^2- (ay)^2)= 2a^2xy/(a^2(x^- y^2)= 2xy/(x^2- y^2) again. That essentially means that the right hand side can be written as a function of y/x. Let v= y/x so that y= xv, y'= xv'+ v and replace y in the right side by xv to get a separable equation for v as a function of x.
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