4.4.50 \(x y'(x)=y(x)-x (x-y(x)) \sqrt {x^2+y(x)^2}\)

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

\[ \left \{ \ln \left (2\,{\frac {x \left (\sqrt {2\,{x}^{2}+2\, \left (y \relax (x ) \right ) ^{2}}+y \relax (x ) +x \right ) }{y \relax (x ) -x}} \right ) +{\frac {\sqrt {2}{x}^{2}}{2}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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