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Math Help - 2nd order parametric differentiation problem

  1. #1
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    2nd order parametric differentiation problem

    I'm slightly confused with this problem:
     x(t) = t^3 + 3t + 1 & y(t) = ln(3t^2 - 3)

    Find the function p(t) such that:

    \frac{d^2y}{dx^2} = p(t)\frac{d^2y}{dt^2}\frac{d^2t}{dx^2}

    I'm new to this subject so any help would be really appreciated.

    Dojo
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by dojo View Post
    I'm slightly confused with this problem:
     x(t) = t^3 + 3t + 1 & y(t) = ln(3t^2 - 3)

    Find the function p(t) such that:

    \frac{d^2y}{dx^2} = p(t)\frac{d^2y}{dt^2}\frac{d^2t}{dx^2}

    I'm new to this subject so any help would be really appreciated.

    Dojo
    First note that

    \displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

    and that

    \displaystyle \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy  }{dx} \right)}{\frac{dx}{dt}}

    Now just compute all of the derivatives and solve for p(t)
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  3. #3
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    Thanks for your help. I end up with this monster! 1/9\, \left( 3\,{t}^{2}+3 \right)  \left( t-1 \right) ^{-1} \left( t+1<br />
 \right) ^{-1} \left( {t}^{2}+1 \right) ^{-1} \left( 6\, \left( 3\,{t}<br />
^{2}-3 \right) ^{-1}-36\,{\frac {{t}^{2}}{ \left( 3\,{t}^{2}-3<br />
 \right) ^{2}}} \right) ^{-1}<br />
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  4. #4
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    is my answer right?
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