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May 5th, 2019, 05:08 AM   #1
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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?
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May 5th, 2019, 05:26 AM   #2
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Originally Posted by psaxno View Post
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 $f'(x)$ ...

$x^2\left[\dfrac{e^x}{x} - \dfrac{e^x(x-1)}{x^2}\right] = xe^x - e^x(x-1) = xe^x - xe^x + e^x = e^x$
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May 5th, 2019, 09:21 AM   #3
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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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