The solution of x d(xy)=f(xy)f1(xy) dx is___

Given : f'(xy)f(xy)d(xy)=dxx∫f'(xy)f(xy)d(xy)=∫1xdx⇒log(f(xy))=logx+logc ⇒f(xy)=cx

The solution of x d(xy)=f(xy)f1(xy) dx is___

Given : f'(xy)f(xy)d(xy)=dxx∫f'(xy)f(xy)d(xy)=∫1xdx⇒log(f(xy))=logx+logc ⇒f(xy)=cx

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The solution of $x d (x y) = (\frac{f (x y)}{f^{1} (x y)}) d x$ is___

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$f (x y) = c$

$y f (x y) = c$

$x f (x y) = c$

$c x = f (x y)$

answer is D.

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Detailed Solution

Given : $\frac{f' (x y)}{f (x y)} d (x y) = \frac{d x}{x}$
$\int \frac{f' (x y)}{f (x y)} d (x y) = \int \frac{1}{x} d x$
$\Rightarrow \log (f (x y)) = \log x + \log c \Rightarrow f (x y) = c x$

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The solution of $x d (x y) = (\frac{f (x y)}{f^{1} (x y)}) d x$ is___