ODE
\[ 6 x y(x) y'(x)+x (x-2 y(x)) y'(x)^2+y(x)^2-2 x y(x)=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous ODE, \(x^n f\left ( \frac {y}{x} , y' \right )=0\)
Mathematica ✓
cpu = 0.337039 (sec), leaf count = 79
\[\left \{\left \{y(x)\to 2 x-\sqrt {x \left (3 x-2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}}\right \},\left \{y(x)\to 2 x+\sqrt {x \left (3 x-2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}}\right \}\right \}\]
Maple ✓
cpu = 0.12 (sec), leaf count = 121
\[\left [y \left (x \right ) = 0, y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-2 \textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{3}+4 \textit {\_a}^{2}+2 \textit {\_a}}+4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} +2 \textit {\_C1} \right ) x, y \left (x \right ) = \RootOf \left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{3}+4 \textit {\_a}^{2}+2 \textit {\_a}}-4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} \right )+2 \textit {\_C1} \right ) x\right ]\] Mathematica raw input
DSolve[-2*x*y[x] + y[x]^2 + 6*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) + 2*x - Sqrt[x*(-2*E^(C[1]/2) + 3*x)]}, {y[x] -> -E^(C[1]/2) + 2*x + Sqrt[x*(-2*E^(C[1]/2) + 3*x)]}}
Maple raw input
dsolve(x*(x-2*y(x))*diff(y(x),x)^2+6*x*y(x)*diff(y(x),x)-2*x*y(x)+y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 0, y(x) = RootOf(-2*ln(x)+Intat((-2*_a^2+(2*_a^3+4*_a^2+2*_a)^(1/2)+4*_a)/_a/(_a^2-4*_a+1),_a = _Z)+2*_C1)*x, y(x) = RootOf(-2*ln(x)-Intat((2*_a^2+(2*_a^3+4*_a^2+2*_a)^(1/2)-4*_a)/_a/(_a^2-4*_a+1),_a = _Z)+2*_C1)*x]