Solutions of the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx$  is

$\frac{xdy-ydx}{\sqrt{x^2+y^2}}=dx\Rightarrow \frac{xdy-ydx}{\frac{x^2}{\sqrt{1+\frac{y^2}{x^2}}}}=\frac{dx}{x}$

$\int \frac{1}{\sqrt{1+{\left(\frac{y}{x}\right)}^2}}d\left(\frac{y}{x}\right)=\int \frac{dx}{x}$

${\sinh }^{-1}\left(y/x\right)=\log x+\log c$

$\log \left|\frac{y}{x}+\sqrt{1+\frac{y^2}{x^2}}\right|=\log \left|cx\right|$

$\frac{y}{x}+\frac{\sqrt{x^2+y^2}}{x}=cx$

$y+\sqrt{x^2+y^2}=c{x}^2$