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Math Help - Solving differential equations using a change of variable

  1. #1
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    Smile Solving differential equations using a change of variable

    My book is telling me

    d/du((e^u)dy/dx)

    =(e^u)dy/dx + (e^u)(d2y/dx2).(dx/du)

    I understand where the first part has come from, but not sure how the second part was derived. Would anyone kindly provide me with an explanation?
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  2. #2
    Like a stone-audioslave ADARSH's Avatar
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    Erghhhhhhh....:p

    Quote Originally Posted by Erghhh View Post
    My book is telling me

    d/du((e^u)dy/dx)

    =(e^u)dy/dx + (e^u)(d2y/dx2).(dx/du)

    I understand where the first part has come from, but not sure how the second part was derived. Would anyone kindly provide me with an explanation?
    This will make it clear

     \frac{d}{du}\{e^u\frac{dy}{dx}\}

    = \frac{dy}{dx}\frac{d}{du}(e^u) +e^u \frac{d}{du}(\frac{dy}{dx})

    Differnetiation of dy/dx wrt u will be given by (using chain rule)

    = \frac{y'}{dx} \cdot \frac{du}{dx}
    ------------------------------------------
    So differnetiation will be

    =\frac{dy}{dx}e^u + e^u \frac{d^2(y)}{dx^2} \cdot \frac{dx}{du}
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