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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 $e^{-x}$ ... $e^{-x} \cdot y' - e^{-x} \cdot y = -\dfrac{1}{x^2}$ $\left(y \cdot e^{-x}\right)' = -\dfrac{1}{x^2}$ $y \cdot e^{-x} = \dfrac{1}{x} + C$ $y(1) = e \implies 1 = 1 + C \implies C = 0$ $y \cdot e^{-x} = \dfrac{1}{x} \implies y = \dfrac{e^x}{x}$ Tags calculus or analysis, derivative, find Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post harryp Algebra 2 September 8th, 2015 08:11 PM mared Trigonometry 3 July 2nd, 2014 08:09 AM kingkos Algebra 5 November 24th, 2012 11:40 PM kevinman Algebra 8 March 8th, 2012 05:53 AM

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