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  
Math Team Joined: Jul 2011 From: Texas Posts: 3,016 Thanks: 1600  Quote:
$x^2\left[\dfrac{e^x}{x}  \dfrac{e^x(x1)}{x^2}\right] = xe^x  e^x(x1) = xe^x  xe^x + e^x = e^x$  
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 
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