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 = 0.137056 (sec), leaf count = 79
\[\left \{\left \{y(x)\to -\sqrt {x \left (3 x-2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}}+2 x\right \},\left \{y(x)\to \sqrt {x \left (3 x-2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}}+2 x\right \}\right \}\]
Maple ✓
cpu = 0.228 (sec), leaf count = 115
\[ \left \{ \ln \left ( x \right ) -{\frac {1}{2}\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ( {{\it \_a}}^{2}-4\,{\it \_a}+1 \right ) } \left ( -2\,{{\it \_a}}^{2}+\sqrt {2}\sqrt {{\it \_a}\, \left ( {\it \_a}+1 \right ) ^{2}}+4\,{\it \_a} \right ) }{d{\it \_a}}}-{\it \_C1}=0,\ln \left ( x \right ) +{\frac {1}{2}\int ^{{\frac {y \left ( x \right ) }{x}}}\!{\frac {1}{{\it \_a}\, \left ( {{\it \_a}}^{2}-4\,{\it \_a}+1 \right ) } \left ( 2\,{{\it \_a}}^{2}+\sqrt {2}\sqrt {{\it \_a}\, \left ( {\it \_a}+1 \right ) ^{2}}-4\,{\it \_a} \right ) }{d{\it \_a}}}-{\it \_C1}=0,y \left ( x \right ) =0 \right \} \] 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),'implicit')
Maple raw output
y(x) = 0, ln(x)-1/2*Intat((-2*_a^2+2^(1/2)*(_a*(_a+1)^2)^(1/2)+4*_a)/_a/(_a^2-4*_a+1),_a = y(x)/x)-_C1 = 0, ln(x)+1/2*Intat((2*_a^2+2^(1/2)*(_a*(_a+1)^2)^(1/2)-4*_a)/_a/(_a^2-4*_a+1),_a = y(x)/x)-_C1 = 0