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Math Help - Solution required

  1. #1
    Member Altair's Avatar
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    Solution required

    Q.Find the differential equation of all the circles that pass through origin.

    A. (x^2 + y^2)y'' = 2(xy' - y)(1 + (y')^2)
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by Altair View Post
    Q.Find the differential equation of all the circles that pass through origin.

    A. (x^2 + y^2)y'' = 2(xy' - y)(1 + (y')^2)
    Well, I have a plan but you'll need to put in the details.

    If the circle (x - h)^2 + (y - k)^2 = r^2 passes through the origin then h^2 + k^2 = r^2.

    Then the equation of the circle becomes

    x^2 - 2xh + y^2 - 2yk = 0 .... (1)

    Differentiate (1) with respect to x:

    2x - 2h + 2y \, y' - 2k \, y'  = 0 .... (2)

    Differentiate (2) with respect to x:

    2 + 2 (y')^2 + 2 y \, y'' - 2k \, y'' = 0 .... (3)

    Clearly k needs to be eliminated from (3).

    (1) and (2) can be solved simultaneously to get k and h in terms of x, y and y'. You want k. Substitute the result into (3).
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  3. #3
    Member Altair's Avatar
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    Need help with substitution. It gets too complicated at the end.
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  4. #4
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    Quote Originally Posted by Altair View Post
    Need help with substitution. It gets too complicated at the end.
    I'll take a closer look when I have a chance, unless someone else beats me to it.
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