ODE
\[ \left (x^2-y(x)^2 \left (x^2+y(x)^2\right )\right ) y'(x)^2-2 x y(x) y'(x)+y(x)^2=0 \] ODE Classification
[`y=_G(x,y')`]
Book solution method
Change of variable
Mathematica ✓
cpu = 2.67154 (sec), leaf count = 88
\[\text {Solve}\left [\frac {\sqrt {x^2+y(x)^2} y(x) \left (\log \left (\frac {x}{\sqrt {x^2+y(x)^2}}+1\right )-\log \left (1-\frac {x}{\sqrt {x^2+y(x)^2}}\right )\right )}{2 x^2 \sqrt {\frac {y(x)^2 \left (x^2+y(x)^2\right )}{x^4}}}+y(x)=c_1,y(x)\right ]\]
Maple ✓
cpu = 3.801 (sec), leaf count = 60
\[[y \left (x \right ) = -i x, y \left (x \right ) = i x, y \left (x \right ) = -\arctanh \left (\RootOf \left (\arctanh \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 \arctanh \left (\textit {\_Z} \right ) \textit {\_C1} \,\textit {\_Z}^{2}+\textit {\_C1}^{2} \textit {\_Z}^{2}+x^{2} \textit {\_Z}^{2}-x^{2}\right )\right )+\textit {\_C1}]\] Mathematica raw input
DSolve[y[x]^2 - 2*x*y[x]*y'[x] + (x^2 - y[x]^2*(x^2 + y[x]^2))*y'[x]^2 == 0,y[x],x]
Mathematica raw output
Solve[y[x] + ((-Log[1 - x/Sqrt[x^2 + y[x]^2]] + Log[1 + x/Sqrt[x^2 + y[x]^2]])*y[x]*Sqrt[x^2 + y[x]^2])/(2*x^2*Sqrt[(y[x]^2*(x^2 + y[x]^2))/x^4]) == C[1], y[x]]
Maple raw input
dsolve((x^2-(x^2+y(x)^2)*y(x)^2)*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)^2 = 0, y(x))
Maple raw output
[y(x) = -I*x, y(x) = I*x, y(x) = -arctanh(RootOf(arctanh(_Z)^2*_Z^2-2*arctanh(_Z)*_C1*_Z^2+_C1^2*_Z^2+x^2*_Z^2-x^2))+_C1]