I have the following differential equation: $$xe^xy'-\dfrac{y}{x}e^x+x\cos (x)=0$$
I tried using substitution, namely:
Let $v=\dfrac{y}{x}$, then $\dfrac{dy}{dx}=\dfrac{dv}{dx}x+v$. Substituting yields:
$$xe^x\left(\dfrac{dv}{dx}x+v\right)-ve^x+x\cos (x)=\dfrac{dv}{dx}x+v-\dfrac{v}{x}+e^{-x}\cos (x)=0$$
But it is still not separable. I also tried the substitution $v=\dfrac{y}{x^2}$. That didn't work either.
For this problem, a substitution should work; honestly, I think I am making a stupid mistake and that's why I am stuck. Could anyone shed me some light on this equation?
I would start by noticing that
\begin{align*} xe^{x}y' - \frac{y}{x}e^{x} + x\cos(x) = 0 & \Longleftrightarrow y' - \frac{y}{x^{2}} + e^{-x}\cos(x) = 0\\\\ & \Longleftrightarrow e^{1/x}y' - \frac{e^{1/x}}{x^{2}}y = -e^{-x + 1/x}\cos(x)\\\\ & \Longleftrightarrow (ye^{1/x})' = -e^{-x + 1/x}\cos(x) \end{align*}
Apparently, this is as far as one can go.
Hopefully this helps!