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 May 5th, 2019, 05:08 AM #1 Newbie   Joined: Jan 2019 From: Europe Posts: 3 Thanks: 0 How to find f(x) It is given that x²(f(x)-f'(x))=e^x for all x>0 and f(1)=e. I have to prove that f(x)=e^x/x. How can I prove that?
May 5th, 2019, 05:26 AM   #2
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 Originally Posted by psaxno It is given that x²(f(x)-f'(x))=e^x for all x>0 and f(1)=e. I have to prove that f(x)=e^x/x. How can I prove that?
substitute $\dfrac{e^x}{x}$ for $f(x)$ and $\dfrac{e^x(x-1)}{x^2}$ for $f'(x)$ ...

$x^2\left[\dfrac{e^x}{x} - \dfrac{e^x(x-1)}{x^2}\right] = xe^x - e^x(x-1) = xe^x - xe^x + e^x = e^x$

 May 5th, 2019, 09:21 AM #3 Math Team     Joined: Jul 2011 From: Texas Posts: 3,016 Thanks: 1600 let $y=f(x)$ $y' -y = -\dfrac{e^x}{x^2}$ integrating factor is $e^{-x}$ ... $e^{-x} \cdot y' - e^{-x} \cdot y = -\dfrac{1}{x^2}$ $\left(y \cdot e^{-x}\right)' = -\dfrac{1}{x^2}$ $y \cdot e^{-x} = \dfrac{1}{x} + C$ $y(1) = e \implies 1 = 1 + C \implies C = 0$ $y \cdot e^{-x} = \dfrac{1}{x} \implies y = \dfrac{e^x}{x}$

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