\[ -\sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.0227448 (sec), leaf count = 13
\[\left \{\left \{y(x)\to x \sinh \left (c_1+\log (x)\right )\right \}\right \}\] ✓ Maple : cpu = 0.065 (sec), leaf count = 27
\[ \left \{ {\frac {1}{{x}^{2}}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}+{\frac {y \left ( x \right ) }{{x}^{2}}}-{\it \_C1}=0 \right \} \]
\[ xy^{\prime }=\sqrt {x^{2}+y^{2}}+y \]
Let \(y=xv\), then \(y^{\prime }=v+xv^{\prime }\) and the above becomes
\begin {align*} x\left ( v+xv^{\prime }\right ) & =\sqrt {x^{2}+\left ( xv\right ) ^{2}}+xv\\ x\left ( v+xv^{\prime }\right ) & =x\sqrt {1+v^{2}}+xv\\ \left ( v+xv^{\prime }\right ) & =\sqrt {1+v^{2}}+v\\ xv^{\prime } & =\sqrt {1+v^{2}} \end {align*}
Separable.
\[ \frac {dv}{\sqrt {1+v^{2}}}=\frac {1}{x}dx \]
Integrating
\begin {align*} \operatorname {arcsinh}\left ( v\right ) & =\ln x+C\\ v & =\sinh \left ( \ln x+C\right ) \end {align*}
Since \(y=xv\) then
\[ y=x\sinh \left ( \ln x+C\right ) \]
Verification
ode:=x*diff(y(x),x)=sqrt(x^2+y(x)^2)+y(x); y0:=x*sinh(ln(x)+_C1); odetest(y(x)=y0,ode) assuming x>= 0; 0