\[ a \sqrt {x^2+y(x)^2}+x y'(x)-y(x)=0 \] ✓ Mathematica : cpu = 0.023155 (sec), leaf count = 16
\[\left \{\left \{y(x)\to x \sinh \left (c_1-a \log (x)\right )\right \}\right \}\]
✓ Maple : cpu = 0.032 (sec), leaf count = 33
\[ \left \{ {\frac {{x}^{a}}{x}\sqrt { \left ( y \left ( x \right ) \right ) ^{2}+{x}^{2}}}+{\frac {{x}^{a}y \left ( x \right ) }{x}}-{\it \_C1}=0 \right \} \]
\[ xy^{\prime }=-a\sqrt {x^{2}+y^{2}}+y \]
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}\left ( v\right ) & =-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
ode:=x*diff(y(x),x)=-a*sqrt(x^2+y(x)^2)+y(x); y0:=x*sinh(_C1-a*ln(x)); odetest(y(x)=y0,ode) assuming x >=0; 0