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  4. The solution of (x+y)2 (x (dy/dx)+y)=xy (1 +(dy/dx)) is

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Q. The solution of $\left(x+y\right)^{2} \left(x \frac{dy}{dx}+y\right)=xy \left(1 +\frac{dy}{dx}\right) $ is

COMEDKCOMEDK 2014Differential Equations

Solution:

Given D.E is
$\left(x+y\right)^{2} \left(x \frac{dy}{dx} +y\right) =xy \left(1 + \frac{dy}{dx}\right) $
$\Rightarrow \left(xy\right)^{-1} \left(x \frac{dy}{dx}+y\right)= \left(x+y\right)^{-2} \left(1+\frac{dy}{dx}\right)$
$\Rightarrow \int\left(xy\right)^{-1} \left(x \frac{dy}{dx} +y\right) = \int \left(x +y\right)^{-2} \left(1 + \frac{dy}{dx} \right)$
Using integral $\int\left(f\left(x\right)\right)^{n} f'\left(x\right)dx = \frac{\left(f\left(x\right)\right)^{n+1}}{n+1} $ and $\int\frac{f'\left(x\right)}{f\left(x\right)} dx = \log\left(f\left(x\right)\right)+C $
$\Rightarrow \log\left(xy\right) =\frac{\left(x+y\right)^{-1}}{-1} +C $
$\Rightarrow \log\left(xy\right) =\frac{-1}{x+y} +C$