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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 

  1. $\ln\left (1+\dfrac{y^{2}}{x^{2}}\right)=x-1$
  2. $\dfrac{1}{2}\ln\left (1+\dfrac{y^{2}}{x^{2}}\right)=x-1$
  3. $\ln\left (1+\dfrac{y}{x}\right)=x-1$
  4. $\dfrac{1}{2}\ln\left (1+\dfrac{y}{x}\right)=x-1$
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