$x^2\frac{dy}{dx}+xy+1=0$ Solve the following differential equation by the form of homogeneous equation. Letting $y=vx$
The equation:
$x^2\frac{dy}{dx}+xy+1=0$
I can’t separate variables
My solution steps are:
$x^2(v+x \frac{dv}{dx})+x^2v+1=0$
$(v+x \frac{dv}{dx})+v=\frac{-1}{x^2}$
$2v+x \frac{dv}{dx}=\frac{-1}{x^2}$
How to separate variables x and v?

Divide the proposed DE by $x$ in order to obtain
\begin{align*}
x^{2}y' + xy + 1 = 0 & \Longleftrightarrow xy' + y = -\frac{1}{x}\\\\
& \Longleftrightarrow (xy)' = -\frac{1}{x}\\\\
& \Longleftrightarrow xy = -\ln|x| + k\\\\
& \Longleftrightarrow y(x) = -\frac{\ln|x|}{x} + \frac{k}{x}
\end{align*}
and we are done.
Hopefully this helps !
