Solution of the equation xdy=(y+xf(yx)f′(yx))dx

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Solution of the equation

x d y = (y + x \frac{f (\frac{y}{x})}{f^{'} (\frac{y}{x})}) d x

Options:

A . f(xy)=cy

B . f(yx)=cx

C . f(yx)=cxy

D . f(yx)=0

Answer: Option B
:
B
We have,

x d y = (y + \frac{x f (\frac{y}{x})}{f^{'} (\frac{y}{x})}) d x

\Rightarrow \frac{d y}{d x} = \frac{y}{x} + \frac{f (\frac{y}{x})}{f^{'} (\frac{y}{x})}

which is homogenous
Putting

y = v x \Rightarrow \frac{d y}{d x} = v + x \frac{d v}{d x}

, we obtain

v + x \frac{d v}{d x} = v + \frac{f (v)}{f^{'} (v)} \Rightarrow \frac{f^{'} (v)}{f (v)} d v = \frac{d x}{x}

Integrating, we get log

f (v) = l o g x + l o g c

\Rightarrow l o g f (v) = l o g c x \Rightarrow f (\frac{y}{x}) = c x

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