ODE
\[ -2 x y(x) y'(x)+x (x-2 y(x)) y'(x)^2+y(x)^2-2 x y(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\), Solve for \(p\)
Mathematica ✓
cpu = 0.279226 (sec), leaf count = 69
\[\left \{\left \{y(x)\to e^{\frac {c_1}{2}}-\sqrt {x \left (2 e^{\frac {c_1}{2}}-x\right )}\right \},\left \{y(x)\to \sqrt {x \left (2 e^{\frac {c_1}{2}}-x\right )}+e^{\frac {c_1}{2}}\right \}\right \}\]
Maple ✓
cpu = 26.886 (sec), leaf count = 103
\[ \left \{ \ln \left ( x \right ) -{\frac {1}{2}\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ( {{\it \_a}}^{2}+1 \right ) } \left ( -2\,{{\it \_a}}^{2}+\sqrt {2}\sqrt {{\it \_a}\, \left ( {\it \_a}-1 \right ) ^{2}} \right ) }{d{\it \_a}}}-{\it \_C1}=0,\ln \left ( x \right ) +{\frac {1}{2}\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ( {{\it \_a}}^{2}+1 \right ) } \left ( 2\,{{\it \_a}}^{2}+\sqrt {2}\sqrt {{\it \_a}\, \left ( {\it \_a}-1 \right ) ^{2}} \right ) }{d{\it \_a}}}-{\it \_C1}=0,y \left ( x \right ) =0 \right \} \] Mathematica raw input
DSolve[-2*x*y[x] + y[x]^2 - 2*x*y[x]*y'[x] + x*(x - 2*y[x])*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^(C[1]/2) - Sqrt[(2*E^(C[1]/2) - x)*x]}, {y[x] -> E^(C[1]/2) + Sqrt[(2*E^(C[1]/2) - x)*x]}}
Maple raw input
dsolve(x*(x-2*y(x))*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-2*x*y(x)+y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = 0, ln(x)-1/2*Intat((-2*_a^2+2^(1/2)*(_a*(_a-1)^2)^(1/2))/_a/(_a^2+1),_a = y(x)/x)-_C1 = 0, ln(x)+1/2*Intat((2*_a^2+2^(1/2)*(_a*(_a-1)^2)^(1/2))/_a/(_a^2+1),_a = y(x)/x)-_C1 = 0