$y d x+x \log \left(\frac{y}{x}\right) d y-2 x d y=0$
$y d x+x \log \left(\frac{y}{x}\right) d y-2 x d y=0$
$\Rightarrow y d x=\left[2 x-x \log \left(\frac{y}{x}\right)\right] d y$
$\Rightarrow \frac{d y}{d x}=\frac{y}{2 x-x \log \left(\frac{y}{x}\right)}$ ...(1)
Let $F(x, y)=\frac{y}{2 x-x \log \left(\frac{y}{x}\right)}$.
$\therefore F(\lambda x, \lambda y)=\frac{\lambda y}{2(\lambda x)-(\lambda x) \log \left(\frac{\lambda y}{\lambda x}\right)}=\frac{y}{2 x-\log \left(\frac{y}{x}\right)}=\lambda^{0} \cdot F(x, y)$
Therefore, the given differential equation is a homogeneous equation.
To solve it, we make the substitution as:
y = vx
$\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(v x)$
$\Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$
Substituting the values of $y$ and $\frac{d y}{d x}$ in equation (1), we get:
$v+x \frac{d v}{d x}=\frac{v x}{2 x-x \log v}$
$\Rightarrow v+x \frac{d v}{d x}=\frac{v}{2-\log v}$
$\Rightarrow x \frac{d v}{d x}=\frac{v}{2-\log v}-v$
$\Rightarrow x \frac{d v}{d x}=\frac{v-2 v+v \log v}{2-\log v}$
$\Rightarrow x \frac{d v}{d x}=\frac{v \log v-v}{2-\log v}$
$\Rightarrow \frac{2-\log v}{v(\log v-1)} d v=\frac{d x}{x}$
$\Rightarrow \frac{2-\log v}{v(\log v-1)} d v=\frac{d x}{x}$
$\Rightarrow\left[\frac{1+(1-\log v)}{v(\log v-1)}\right] d v=\frac{d x}{x}$
$\Rightarrow\left[\frac{1}{v(\log v-1)}-\frac{1}{v}\right] d v=\frac{d x}{x}$
Integrating both sides, we get:
$\int \frac{1}{v(\log v-1)} d v-\int \frac{1}{v} d v=\int \frac{1}{x} d x$
$\Rightarrow \int \frac{d v}{v(\log v-1)}-\log v=\log x+\log \mathrm{C}$ $\ldots(2)$
$\Rightarrow$ Let $\log v-1=t$
$\Rightarrow \frac{d}{d v}(\log v-1)=\frac{d t}{d v}$
$\Rightarrow \frac{1}{v}=\frac{d t}{d v}$
$\Rightarrow \frac{d v}{v}=d t$
Therefore, equation (1) becomes:
$\Rightarrow \int \frac{d t}{t}-\log v=\log x+\log \mathrm{C}$
$\Rightarrow \log t-\log \left(\frac{y}{x}\right)=\log (\mathrm{C} x)$
$\Rightarrow \log \left[\log \left(\frac{y}{x}\right)-1\right]-\log \left(\frac{y}{x}\right)=\log (\mathrm{C} x)$
$\Rightarrow \log \left[\frac{\log \left(\frac{y}{x}\right)-1}{\frac{y}{x}}\right]=\log (\mathrm{C} x)$
$\Rightarrow \frac{x}{y}\left[\log \left(\frac{y}{x}\right)-1\right]=\mathrm{C} x$
$\Rightarrow \log \left(\frac{y}{x}\right)-1=\mathrm{C} y$
This is the required solution of the given differential equation.
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