ODE
\[ x y'(x)=y(x)-x (x-y(x)) \sqrt {x^2+y(x)^2} \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Book solution method
Homogeneous equation, \(xy'(x)=x f(x) g(u)+y(x)\)
Mathematica ✓
cpu = 0.115472 (sec), leaf count = 81
\[\left \{\left \{y(x)\to \frac {x \left (-2 e^{\frac {2 c_1+x^2}{\sqrt {2}}}+e^{\sqrt {2} \left (2 c_1+x^2\right )}-1\right )}{2 e^{\frac {2 c_1+x^2}{\sqrt {2}}}+e^{\sqrt {2} \left (2 c_1+x^2\right )}-1}\right \}\right \}\]
Maple ✓
cpu = 0.252 (sec), leaf count = 49
\[ \left \{ \ln \left ( 2\,{\frac {x \left ( \sqrt {2\,{x}^{2}+2\, \left ( y \left ( x \right ) \right ) ^{2}}+y \left ( x \right ) +x \right ) }{y \left ( x \right ) -x}} \right ) +{\frac {\sqrt {2}{x}^{2}}{2}}-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*y'[x] == y[x] - x*(x - y[x])*Sqrt[x^2 + y[x]^2],y[x],x]
Mathematica raw output
{{y[x] -> ((-1 - 2*E^((x^2 + 2*C[1])/Sqrt[2]) + E^(Sqrt[2]*(x^2 + 2*C[1])))*x)/(-1 + 2*E^((x^2 + 2*C[1])/Sqrt[2]) + E^(Sqrt[2]*(x^2 + 2*C[1])))}}
Maple raw input
dsolve(x*diff(y(x),x) = y(x)-x*(x-y(x))*(x^2+y(x)^2)^(1/2), y(x),'implicit')
Maple raw output
ln(2*x*((2*x^2+2*y(x)^2)^(1/2)+y(x)+x)/(y(x)-x))+1/2*2^(1/2)*x^2-ln(x)-_C1 = 0