JEE Math Practice Question

If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2 \mathrm{e}^{\tan ^{-1} y}\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :

1. A.$\mathrm{e}^{\pi / 4}$
2. B.$e^{\pi / 12}$
3. C.$\mathrm{e}^{\pi / 6}$
4. D.$e^{\pi / 3}$