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Math Help - Transformation

  1. #1
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    Wink Transformation

    (2x)d2t/dx2 - 6(dx/dt)^2 = x^2 - 3x^4


    show that the substitution y=1/x^2 transforms this differential eqn into d2y/dt2 + y = 3

    So i've work out

    -2/x^3(d2x/dt2) + 6/x^4(dx/dt)

    i think the latter part of this eqn is wrong as the mark scheme says it's 6/x^4(dx/dt)^2

    Can anyone explain this?
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  2. #2
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    Quote Originally Posted by Erghhh View Post
    (2x)d2t/dx2 - 6(dx/dt)^2 = x^2 - 3x^4


    show that the substitution y=1/x^2 transforms this differential eqn into d2y/dt2 + y = 3

    So i've work out

    -2/x^3(d2x/dt2) + 6/x^4(dx/dt)

    i think the latter part of this eqn is wrong as the mark scheme says it's 6/x^4(dx/dt)^2

    Can anyone explain this?
    Sure

    y = x^{-2} so \frac{dy}{dt} = -2x^{-3} \frac{dx}{dt} . Thus,
    \frac{d^2y}{dt^2} = \frac{d}{dt} \left(-2x^{-3} \frac{dx}{dt}\right) = 6x^{-4} \underbrace{\frac{dx}{dt} \cdot \frac{dx}{dt}}_{ \dot{x}^2 } -2x^{-3} \frac{d^2x}{dt^2}
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