1) ${y}^{\prime}(x)=\frac{x-y(x)}{x+y(x)}$ with the initial condition $y(1)=1$ .I'm arrived to prove that

$y=x(\sqrt{2-{e}^{-2(\mathrm{ln}x+c)}}-1)$

but I don't know if it's correct. If it's right, how do I find the constant $c$? Because WolframAlpha says that the solution is $y(x)=\sqrt{2}\sqrt{{x}^{2}+1}-x$.

2) ${y}^{\prime}(x)=\frac{2y(x)-x}{2x-y(x)}$. I'm arrived to prove that $\frac{z-1}{(z+1{)}^{3}}={e}^{2c}{x}^{2}$ but I don't know if it's correct. If it's right, how do I explain $z$ to substitute it in $y=xz$? Then, how do I find the constant $c$ ?