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Math Help - ODEs differentiating y with respect to x

  1. #1
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    ODEs differentiating y with respect to x

    What is the derivative with respect to x of . . . . xy'/y^2
    I can get as far as derivative of xy' = y' + xy"
    But I dont know what to do about the bit on the bottom . .
    I'm guessing i need the quotient rule somehow . .

    Cheers
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  2. #2
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    Quote Originally Posted by eatmoreburritos View Post
    What is the derivative with respect to x of . . . . xy'/y^2
    I can get as far as derivative of xy' = y' + xy"
    But I dont know what to do about the bit on the bottom . .
    I'm guessing i need the quotient rule somehow . .

    Cheers
    You have to use the quotient rule: The derivative of \frac{u}{v} is equal to  \frac{\frac{du}{dx} \, v - \frac{dv}{dx} \, u}{v^2}.

    In your problem:

    u = xy' \Rightarrow \frac{du}{dx} = y' + x \, y'', as you correctly found (using the product rule).

    v = y^2. To get \frac{dv}{dx} you have to use the chain rule: \frac{dv}{dx} = 2y\, y'.
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