The solution of the differential equation \(\left(3 x y+y^{2}\right) d x+\left(x^{2}+x y\right) d y=0\) is

Homogeneous equation can be written in the form of
Put
\(
\begin{aligned}
\frac{d y}{d x} &=-\frac{3 x y+y^{2}}{x^{2}+x y} \\
y=& v x \text { and } \frac{d y}{d x}=v+x \frac{d v}{d x}, \text { we get }
\end{aligned}
\)
\(
v+x \frac{d v}{d x}=-\frac{3 x^{2} v+x^{2} v^{2}}{x^{2}+x^{2} v}
\)
\(
\Rightarrow \quad x \frac{d v}{d x}=\frac{-2 v(v+2)}{v+1}
\)
\(\Rightarrow \frac{1}{x} d x=-\frac{(v+1)}{2 v(v+2)} d v \)
\( \Rightarrow -\frac{2}{x}=-\left[\frac{1}{2(v+2)}+\frac{1}{2 v}\right] d v\)
On integrating, we get
\(-2 \log _{e} x=\frac{1}{2} \log (v+2)+\frac{1}{2} \log v-\log c \)
\( \Rightarrow v(v+2) x^{4}=c^{2} \)
\( \Rightarrow \frac{y}{x}\left(\frac{y}{x}+2\right) x^{4}=c^{2} \left(\because v=\frac{y}{x}\right)\)
Hence, required solution is \(\left(y^{2}+2 x y\right) x^{2}=c^{2}\)