Answer
$y=\sqrt {2\ln|x|+1}$
Work Step by Step
We are given that $xy \dfrac{dy}{dx}=1$
We will separate the variables to obtain:
$y \ dy=\dfrac{dx}{x}$
Integrate to obtain:
$\int y \ dy=\int\dfrac{dx}{x}$
This implies that $\dfrac{y^2}{2}=\ln|x|+C$
After applying the initial conditions, $y=1$ when $x=1$, we get $C=\dfrac{1}{2}$
Therefore, we have: $\dfrac{y^2}{2}=\ln|x|+\dfrac{1}{2} \implies y=\sqrt {2\ln|x|+1}$
