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Math Help - If x=e^t, what is d^2 y/dx^2 ?

  1. #1
    Member ssadi's Avatar
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    If x=e^t, what is d^2 y/dx^2 ?

    Given x=e^t, what is the expression for \frac{d^2 y}{dx^2}?

    I need it to solve a second order differential equation.

    I get \frac{dy}{dx}=e^{-t} \frac{dy}{dt}, but when i tried to differentiate the expression e^{-t} \frac{dy}{dt} i am stuck at d/dx (dy/dt), help me out please.
    Last edited by mr fantastic; February 3rd 2009 at 07:12 AM. Reason: Fixed the latex
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  2. #2
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    Quote Originally Posted by ssadi View Post
    Given x=e^t, what is the expression for \frac{d^2 y}{dx^2}?
    I need it to solve a second order differential equation.
    I get \frac{dy}{dx}=e^{-t} \frac{dy}{dt}, but when i tried to differentiate the expression e^{-t} \frac{dy}{dt} i am stuck at d/dx (dy/dt), help me out please.
    \frac{d^2 y}{dx^2} = \frac{d}{dx} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[\frac{dy}{dx} \right] \cdot \frac{dt}{dx}.

    Now note that \frac{d}{dt} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[e^{-t} \cdot \frac{dy}{dt}\right] = - e^{-t} \frac{dy}{dt} + \frac{d^2 y}{dt^2} e^{-t}.
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  3. #3
    Member ssadi's Avatar
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    Quote Originally Posted by mr fantastic View Post
    \frac{d^2 y}{dx^2} = \frac{d}{dx} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[\frac{dy}{dx} \right] \cdot \frac{dt}{dx}.

    Now note that \frac{d}{dt} \left[\frac{dy}{dx} \right] = \frac{d}{dt} \left[e^{-t} \cdot \frac{dy}{dt}\right] = - e^{-t} \frac{dy}{dt} + \frac{d^2 y}{dt^2} e^{-t}.
    Ok, got the required form. Thanks for the helping hand
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