Solution of xdy=f(xy)xf1(xy)−ydx is

xdy+ydx=f(xy)xf1(xy)dx ⇒d(xy)=f(xy)xf1(xy)dx ⇒f1(xy)f(xy)d(xy)=dxx ⇒∫f1(xy)f(xy)d(xy)=∫dxx ⇒log fxy=log x+logc ⇒fxy=xc

Solution of xdy=f(xy)xf1(xy)−ydx is

xdy+ydx=f(xy)xf1(xy)dx ⇒d(xy)=f(xy)xf1(xy)dx ⇒f1(xy)f(xy)d(xy)=dxx ⇒∫f1(xy)f(xy)d(xy)=∫dxx ⇒log fxy=log x+logc ⇒fxy=xc

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Solution of $xdy = (\frac{f (xy)}{{xf}^{1} (xy)} - y) dx$ is

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$f (xy) = c$

$xf (xy) = c$

$yf (xy) = c$

$cx = f (xy)$

answer is D.

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Detailed Solution

$xdy + ydx = \frac{f (xy)}{{xf}^{1} (xy)} dx \Rightarrow d (xy) = \frac{f (xy)}{{xf}^{1} (xy)} dx \Rightarrow \frac{f^{1} (xy)}{f (xy)} d (xy) = \frac{dx}{x} \Rightarrow \int \frac{f^{1} (xy)}{f (xy)} d (xy) = \int \frac{dx}{x} \Rightarrow \log f (x y) = \log x + \log c \Rightarrow f (x y) = x c$