\[ a \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.117947 (sec), leaf count = 16
\[\{\{y(x)\to x \sinh (-a \log (x)+c_1)\}\}\] ✓ Maple : cpu = 0.039 (sec), leaf count = 33
\[\frac {x^{a} \sqrt {y \relax (x )^{2}+x^{2}}}{x}+\frac {x^{a} y \relax (x )}{x}-c_{1} = 0\]
Hand solution
\[ xy^{\prime }=-a\sqrt {x^{2}+y^{2}}+y \]
Let \(y=xv\), then \(y^{\prime }=v+xv^{\prime }\) and the above becomes
\begin {align*} x\left (v+xv^{\prime }\right ) & =-a\sqrt {x^{2}+\left (xv\right ) ^{2}}+xv\\ x\left (v+xv^{\prime }\right ) & =-ax\sqrt {1+v^{2}}+xv\\ \left (v+xv^{\prime }\right ) & =-a\sqrt {1+v^{2}}+v\\ xv^{\prime } & =-a\sqrt {1+v^{2}} \end {align*}
Separable.
\[ \frac {dv}{\sqrt {1+v^{2}}}=\frac {-a}{x}dx \]
Integrating
\begin {align*} \operatorname {arcsinh}\relax (v) & =-a\ln x+C\\ v & =\sinh \left (C-a\ln x\right ) \end {align*}
Since \(y=xv\) then
\[ y=x\sinh \left (C-a\ln x\right ) \]
Verification