$$(y - \frac{1}{x})dx + \frac{1}{y}dy = 0 \,\,\, , \,\,\, y(\sqrt[]{2}) = \sqrt[]{2}$$
So I need to find the integrating factor. $\frac{M_y - N_x}{N} = \frac{1 - 0}{0}$ which we cannot have so that leaves us with only $\frac{N_x - M_y}{M} = -1$. This last condition says if the result is dependent on $y$, then the integrating factor is $u(y) = exp(\int \frac{N_x - M_y}{M})$, but our result was a constant. It relies neither on $x$ or $y$. What now?