<s>[INST] Integrating Factor and the Exact Equation $(y-\frac{1}{x})dx+\frac{1}{y}dy=0$. 
$$(y - \frac{1}{x})dx + \frac{1}{y}dy = 0 \,\,\, , \,\,\, y(\sqrt[]{2}) = \sqrt[]{2}$$

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?
 [/INST] We can get an exact form
$$\left(y - \frac{1}{x} \right)dx + \frac{1}{y}dy = 0 \implies ydx-xdy=xy^2 dx$$
$$\frac{ydx-xdy}{y^2}=xdx \implies d\left(\frac{x}{y} \right)=xdx$$
On integration we get
$$\frac{x}{y} =\frac{x^2}{2} +k$$
$k$ is some constant
Now using $y(\sqrt{2})=\sqrt{2}$ we get $k=0$
Therefore the specific solution is $y=\frac{2}{x}$
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