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All in the family

  1. Oct 27, 2013 #1
    1. The problem statement, all variables and given/known data
    For the family of curves given by (x^2)/(a^2) + (y^2)/(b^2) = 1, find the family of curves that are orthogonal to these.


    2. Relevant equations
    Here I will number my steps
    1. Differentiate
    2. opposite reciprocal
    3. integrate
    4. simplify



    3. The attempt at a solution

    1. dy/dx = -(xb^2)/(ya^2) this what I have for my derivative
    2. dy/dx = (ya^2)/(xb^2 ) flip and sign.
    3. After integration I have (1/a^2) lny = (1/b^2)lnx
    4. I just tried to simplify it

    So from
    (1/a^2) lny = (1/b^2)lnx
    ln(y) = (a^2/b^2)ln(x)
    I just got ride of ln with e.
    y = x^(a^2/b^2) but I feel like I should have c in my equation immediately after integration. So rather
    just left as it was ?
     
  2. jcsd
  3. Oct 27, 2013 #2
    You're nearly there. The C arises during integration -- you should get ## log y = \frac{a^2}{b^2}logx + C.## When you exponentiate you get ##y = x^{ \frac{a^2}{b^2}} * e^C. ## Of course ##e^C## is just some constant.

    One thing I would suggest is -- don't call 2 different things dy/dx. It's passable here because the problem is very simple. But if you had something more complex, and you discover on page 6 that you made an error on page 3, and then you can't figure out what dy/dx is supposed to be -- that is a mess. If in the equation for the orthogonal family you change the y to w, and write dw/dx ... then all is clear.
     
  4. Oct 27, 2013 #3
    Good advice thx.
     
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