Let $F(x,y)=x \hat{i}+ y\hat{j}$. Its field lines satisfy $$\cfrac{dx}{x}=\cfrac{dy}{y}.$$ Then, I write $ydx = xdy$ so that $xy +C_1 = xy +C_2 $ which implies $xy=C$ for some constant $C$. This is how its found in the examples. However, the book says that $\cfrac{dx}{x}=\cfrac{dy}{y} \implies \ln(x)+C_1 = \ln(y)+C_2 \implies \ln(y) = \ln(x) +C \implies y =Dx$.
So, what is the argument that I am missing? Thanks.