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 = 0.448409 (sec), leaf count = 149
\[\left \{\left \{y(x)\to \frac {x-8 \sqrt {x^2 \sinh ^6\left (\frac {x^2+2 c_1}{\sqrt {2}}\right ) \text {csch}^4\left (\sqrt {2} \left (x^2+2 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {x^2+2 c_1}{\sqrt {2}}\right )}\right \},\left \{y(x)\to \frac {x+8 \sqrt {x^2 \sinh ^6\left (\frac {x^2+2 c_1}{\sqrt {2}}\right ) \text {csch}^4\left (\sqrt {2} \left (x^2+2 c_1\right )\right )}}{-1+2 \tanh ^2\left (\frac {x^2+2 c_1}{\sqrt {2}}\right )}\right \}\right \}\]
Maple ✓
cpu = 0.219 (sec), leaf count = 49
\[\left [\ln \left (\frac {2 x \left (\sqrt {2 x^{2}+2 y \left (x \right )^{2}}+y \left (x \right )+x \right )}{-x +y \left (x \right )}\right )+\frac {\sqrt {2}\, x^{2}}{2}-\ln \left (x \right )-\textit {\_C1} = 0\right ]\] 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] -> (x - 8*Sqrt[x^2*Csch[Sqrt[2]*(x^2 + 2*C[1])]^4*Sinh[(x^2 + 2*C[1])/Sqrt[2]]^6])/(-1 + 2*Tanh[(x^2 + 2*C[1])/Sqrt[2]]^2)}, {y[x] -> (x + 8*Sqrt[x^2*Csch[Sqrt[2]*(x^2 + 2*C[1])]^4*Sinh[(x^2 + 2*C[1])/Sqrt[2]]^6])/(-1 + 2*Tanh[(x^2 + 2*C[1])/Sqrt[2]]^2)}}
Maple raw input
dsolve(x*diff(y(x),x) = y(x)-x*(x-y(x))*(x^2+y(x)^2)^(1/2), y(x))
Maple raw output
[ln(2*x*((2*x^2+2*y(x)^2)^(1/2)+y(x)+x)/(-x+y(x)))+1/2*2^(1/2)*x^2-ln(x)-_C1 = 0]