$xdy - ydx = \sqrt {{x^2} + {y^2}} d\kappa$

: $xdy - ydx = \sqrt {{x^2} + {y^2}} dx$,

which can be written as
$x\cfrac{{dy}}{{dx}} = y + \sqrt {{x^2} + {y^2}} \Rightarrow \cfrac{{dy}}{{dx}} = \cfrac{y}{x} + \sqrt {1 + {{\left( {\cfrac{y}{x}} \right)}^2}}$

…(1)
Since R.H.S. is of the form $g(y/x)$ ,

so it is a homogeneous function of degree zero.

Therefore, equation (1) is a homogeneous differential equation.

To solve this, we put $y = vx \Rightarrow \cfrac{{dy}}{{dx}} = v + x\cfrac{{dv}}{{dx}}$

Substituting the values of $y$and $\cfrac{{dy}}{{dx}}$ in (1),

we get
$v + x\cfrac{{dv}}{{dx}} = v + \sqrt {1 + {v^2}} \Rightarrow x\cfrac{{dv}}{{dx}} = \sqrt {1 + {v^2}}$

$\Rightarrow \cfrac{{dx}}{x} = \cfrac{{dv}}{{\sqrt {1 + {v^2}} }} \Rightarrow \int {\cfrac{{dx}}{x}} = \int {\cfrac{{dv}}{{\sqrt {1 + {v^2}} }}}$

[Integrating both sides]
$\Rightarrow \log x + \log {C_1} = \log \left| {\cfrac{y}{x} + \sqrt {1 + \cfrac{{{y^2}}}{{{x^2}}}} } \right|$
$\Rightarrow \log {C_1}x = \log |y + \sqrt {{x^2} + {y^2}} | - \log x$

$\Rightarrow \pm {C_1}{x^2} = y + \sqrt {{x^2} + {y^2}} \Rightarrow C{x^2} = y + \sqrt {{x^2} + {y^2}} \,\,\,\,\left[ {C = \pm {C_1}} \right]$

which is required general solution.