Discussion Forum

If $x \frac{dy}{dx}+y=x.\frac{f(xy)}{f'(xy)}$, then f(xy) is equal to 

Answer:

Explanation:

$x \frac{dy}{dx}+y=x.\frac{f(xy)}{f'(xy)}$,

i.e, $\frac{d}{dx}(xy)=x \frac{f(x,y)}{f'(x,y)}$

$\Rightarrow$        $\frac{f'(xy)}{f(xy)} d(xy)=x dx$

$\Rightarrow$        $\int \frac{f'(xy)}{f(xy)} d(xy)=\int xdx$

$\Rightarrow$   $\log [f(xy)]= \frac{x^{2}}{2}+C$

$\Rightarrow$      $f(xy)= e^{(x^{2}/2+C)}$

$\Rightarrow$    $e^{x^{2}/2} .e^{C}$= $k.e^{\frac{x^{2}}{2}}$