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Math Help - now i have problem that cant be proofed

  1. #1
    Junior Member
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    now i have problem that cant be proofed

    if Y=[X(1-X)]



    prove that: Y[second derivative]+[first derivative]+1=0



    i think there is something wroung in it
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, mohamedsafy!

    It took me a few tries, but I got it . . .


    \text{If }\,y \:=\:\left(x-x^2\right)^{\frac{1}{2}} .[1]

    prove that: . y\left(\frac{d^2y}{dx^2}\right)+ \left(\frac{dy}{dx}\right)^2 +1 \:=\:0
    First, we find the derivatives . . .


    \frac{dy}{dx} \;=\;\tfrac{1}{2}(x - x^2)^{-\frac{1}{2}}(1-2x) \quad\Rightarrow\quad\frac{dy}{dx}\;=\;\frac{1-2x}{2\sqrt{x-x^2}} .[2]


    \frac{d^2y}{dx^2} \;=\;\frac{2\sqrt{x-x^2}(-2) - (1-2x)\cdot2\cdot\frac{1}{2}(x-x^2)^{-\frac{1}{2}}(1-2x)}{4(x-x^2)}

    . . which simplifies to: . \frac{d^2y}{dx^2} \;=\;\frac{-1}{4(x-x^2)^{\frac{3}{2}}} .[3]


    Then: .. . y\;\;\cdot\;\;\left(\frac{d^2y}{dx^2}\right) \quad+ \quad\;\;\left(\frac{dy}{dx}\right)^2\;\; +\;\; 1 . .
    Substitute [1], [2] and [3]

    . . = \;\overbrace{\sqrt{x-x^2}}\cdot\overbrace{\frac{-1}{4(x-x^2)^{\frac{3}{2}}}} + \overbrace{\left(\frac{1-2x}{2\sqrt{x-x^2}}\right)^2\,} + 1

    . . = \;\frac{-1}{4(x-x^2)} + \frac{(1-2x)^2}{4(x-x^2} + \frac{4(x-x^2)}{4(x-x^2)} \;=\;\frac{-1 + (1-2x)^2 + 4(x-x^2)}{4(x-x^2)}

    . . = \;\frac{-1 + 1 - 4x + 4x^2 + 4x - 4x^2}{4(x-x^2)} \;=\;\frac{0}{4(x-x^2)} \;=\;0\quad\hdots There!

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  3. #3
    Junior Member
    Joined
    Nov 2008
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    oh thxxxxx

    thank u now i knew where was my error
    thankx for all people in this forum u r really helping me too much to pass my exam tommorow i really apreciae it
    Last edited by mr fantastic; December 27th 2008 at 05:42 AM. Reason: Deleted excessive use of smilies.
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