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

  1. #1
    Member Vinod's Avatar
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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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  2. #2
    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 $\displaystyle (x-h)^2+(y-k)^2=a^2$ we get $\displaystyle 2(x-h)+2(y-k)y'=0$ or equivalently $\displaystyle \boxed{(x-h)dx+(y-k)dy=0}$ (this is the solution)
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  3. #3
    Member Vinod's Avatar
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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 $\displaystyle (x-h)^2+(y-k)^2=a^2$ we get $\displaystyle 2(x-h)+2(y-k)y'=0$ or equivalently $\displaystyle \boxed{(x-h)dx+(y-k)dy=0}$ (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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  4. #4
    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 $\displaystyle h,k$ 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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  5. #5
    Member Vinod's Avatar
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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 $\displaystyle h,k$ 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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  6. #6
    Member Vinod's Avatar
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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
    $\displaystyle \frac{dy}{dx}=\frac{-(x-h)}{(y-k)}$

    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; Dec 6th 2011 at 04:44 AM.
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