= Solution
Write the equation as $P\,dx+Q\,dy=0$, with $P=3x+y$ and $Q=2x$. Then
$$
\frac{P_y-Q_x}{Q}=\frac{1-2}{2x}=-\frac1{2x},
$$
so a suitable <integrating factor> is $\mu=|x|^{-1/2}$ on either half-line. For example, on $x>0$ the resulting exact form is
$$
\left(3\sqrt x+\frac y{\sqrt x}\right)dx+2\sqrt x\,dy
=d\left[2\sqrt x(x+y)\right].
$$
Hence the solution, written to cover either sign of $x$, is
$$
\boxed{y=-x+\frac{C}{\sqrt{|x|}},\qquad x\ne0}.
$$
The original equation remains meaningful at $x=0$, where it requires $y=0$. Only $C=0$ gives a finite differentiable extension across zero, namely $y=-x$.
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