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Homework Help: Need help on Orthogonal Trajectories in my Diff. EQ. Class

  1. Feb 5, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that the families (x+c1)(x2+y2)+x = 0 and (y+c2)(x2+y2)-y = 0

    2. Relevant equations
    For the 2 curves to be orthogonal their slopes should be negative recriprocles.

    3. The attempt at a solution

    I'm pretty sure that for the first set of curves:

    y'(x) = - (2c1 x + 3x2+y[x]2+1)/(2(c1+x)y[x])

    and for the second set of curves:

    y'(x) = (2x (c2 +y)) / (2c2 y[x] + x2 + 3y[x]2-1)

    which are not negative recriprocles of each other.

    I'm thinking i went wrong somewhere along the lines of finding the deravitive. if anyone could please help me out i'd really appreciate it.

    I used Mathematica to get those answers:

    For the first set i factored out the original problem then took the deravitive of that:

    D[c1 x^2 + c1 y[x]^2 + x^3 + x y[x]^2 + x, x]

    then i used the solve command to solve that for y'[x]

    For the second set i factored out the original problem, then took the deravitive of that with respect to x:

    D[c2 x^2 + c2 y[x]^2 + x^2 y[x] + y[x]^3 - y[x], x]

    Then i used the Solve[] function to solve that for y'[x]

    P.S. I'm pretty sure these are supposed to be orthogonal just because there isn't an option for not orthogonal.

    Like i said any help would be appreciated.
  2. jcsd
  3. Feb 6, 2010 #2


    Staff: Mentor

    I get almost the same as you did, with the difference being a sign in dy/dx for the second curve.

    For that curve, I got y' = -(2xy + 2c2x)/(x2 + 3y2 + 2c2y - 1)

    I believe that the reason our answers differ is because you weren't working with the equations, which you need to do to differentiate implicitly.

    In any case, I didn't find that the derivatives were negative reciprocals of each other, either, so are you sure you copied the problem correctly?
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