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Math Help - find out differential equation of family of circles represented by(x-h)^2+(y-k)^2=a^2

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    find out differential equation of family of circles represented by(x-h)^2+(y-k)^2=a^2

    Differentiating w.r.t.x we have

    2x-2+2y dy/dx-2dy/dx =0 So,x - 1+ydy/dx-dy/dx =0

    So x-1+(y-1)dy/dx=0

    dy/dx=1-x/y-1

    differentiating again w.r.t.x,we get

    1+(y-1)dy/dx +(dy/dx)^2=0

    Now how to calculate required differential equation.

    If my calculations of 1st and 2nd order derivative is wrong, reply me accordingly.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: find out differential equation of family of circles represented by(x-h)^2+(y-k)^2

    Differentiating both sides of (x-h)^2+(y-k)^2=a^2 we get 2(x-h)+2(y-k)y'=0 or equivalently \boxed{(x-h)dx+(y-k)dy=0} (this is the solution)
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    Post Re: find out differential equation of family of circles represented by(x-h)^2+(y-k)^2

    Quote Originally Posted by FernandoRevilla View Post
    Differentiating both sides of (x-h)^2+(y-k)^2=a^2 we get 2(x-h)+2(y-k)y'=0 or equivalently \boxed{(x-h)dx+(y-k)dy=0} (this is the solution)
    Sir,
    I want the solution without arbitrary constants h,k. So I want to eliminate them by differentiation.
    Okay,
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    MHF Contributor FernandoRevilla's Avatar
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    Re: find out differential equation of family of circles represented by(x-h)^2+(y-k)^2

    Quote Originally Posted by Vinod View Post
    Sir, I want the solution without arbitrary constants h,k. So I want to eliminate them by differentiation. Okay,
    Why do you know that h,k are arbitrary constants? Does the problem mention it? If this is the case, the problem should be stated: Find the differential equation of all circles in the plane.
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    Post Re: find out differential equation of family of circles represented by(x-h)^2+(y-k)^2

    Quote Originally Posted by FernandoRevilla View Post
    Why do you know that h,k are arbitrary constants? Does the problem mention it? If this is the case, the problem should be stated: Find the differential equation of all circles in the plane.
    Sir,
    You are correct. Original problem is as follows:
    Find the differential equation of all circles having a fixed radius a.

    Answer given in the study material is as below:
    {1+ (dy/dx)^2}^3=a^2{dy/dx}^2

    Hence give me a hint to arrive at the above answer.

    Regards
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    Post Re: find out differential equation of family of circles represented by(x-h)^2+(y-k)^2

    Sir,
    I found the differential equation of all circles having fixed radius a.First derivative of equation of these circles is
    \frac{dy}{dx}=\frac{-(x-h)}{(y-k)}

    Second derivative of equation of these circles is
    dy/dx=[-(dy/dx)^2-1]/(y-k)

    Differential equation of all circles having fixed radius a is given in post number #5 to this thread.

    However thanks for your reply.
    Last edited by Vinod; December 6th 2011 at 04:44 AM.
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