\[ -x \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.109527 (sec), leaf count = 12
\[\{\{y(x)\to x \sinh (x+c_1)\}\}\] ✓ Maple : cpu = 3.156 (sec), leaf count = 28
\[\ln \left (\sqrt {y \relax (x )^{2}+x^{2}}+y \relax (x )\right )-x -\ln \relax (x )-c_{1} = 0\]
Hand solution
\[ xy^{\prime }=x\sqrt {x^{2}+y^{2}}+y \]
Let \(y=xv\), then \(y^{\prime }=v+xv^{\prime }\) and the above becomes
\begin {align*} x\left (v+xv^{\prime }\right ) & =x\sqrt {x^{2}+\left (xv\right ) ^{2}}+xv\\ \left (v+xv^{\prime }\right ) & =x\sqrt {1+v^{2}}+v\\ xv^{\prime } & =x\sqrt {1+v^{2}}\\ v^{\prime } & =\sqrt {1+v^{2}} \end {align*}
Separable.
\[ \frac {dv}{\sqrt {1+v^{2}}}=dx \]
Integrating
\begin {align*} \operatorname {arcsinh}\relax (v) & =x+C\\ v & =\sinh \left (x+C\right ) \end {align*}
Since \(y=xv\) then
\[ y=x\sinh \left (x+C\right ) \]
Verification