# The integrating factor of the differential equation $(xy-1)\frac{dy}{dx}+y^{2}=0$ is Option 1) y Option 2) $\frac{1}{y}$ Option 3) $\frac{1}{xy}$ Option 4) xy

Bernoulli's Equation -

$\frac{dy}{dx}+py =Qy^{n}$

- wherein

P,Q are the function of x alone.

$(xy-1)\frac{dy}{dx}+y^{2}=0$

$\therefore- \frac{1}{y^{2}}\frac{dy}{dx}=\frac{1}{xy-1}$

$put\: \frac{1}{y}=\frac{1}{t}$

$\therefore- \frac{1}{y^{2}}\frac{dy}{dx}=\frac{dt}{dx}$

$\frac{dt}{dx}=\frac{1}{\frac{x}{t}-1}$

$\left [ \because y=\frac{1}{t} \right ]$

$\frac{dx}{dt}=\frac{x}{t}-1$

$\frac{dx}{dt}-\frac{x}{t}=-1$

$p=-\frac{1}{t}$

$\therefore \int Pdt=-\int \frac{1}{t}dt$

$=-logt$

$=log\frac{1}{t}$

$I.f=e^{log\frac{1}{t}}$

$=>\frac{1}{t}=y$

Option 1)

y

Option is correct

Option 2)

$\frac{1}{y}$

Option is incorrect

Option 3)

$\frac{1}{xy}$

Option is incorrect

Option 4)

xy

Option is incorrect

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