A curve passes through the point $\left ( x=1,y=0 \right )$ and satisfies the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x^{2}+y^{2}}{2y}+\dfrac{y}{x}.$ The equation that describes the curve is
- $\ln\left (1+\dfrac{y^{2}}{x^{2}}\right)=x-1$
- $\dfrac{1}{2}\ln\left (1+\dfrac{y^{2}}{x^{2}}\right)=x-1$
- $\ln\left (1+\dfrac{y}{x}\right)=x-1$
- $\dfrac{1}{2}\ln\left (1+\dfrac{y}{x}\right)=x-1$