ODE
\[ -2 x \left (x^3+2 y(x)^2\right ) y(x) y'(x)+\left (2 x^3+y(x)^2\right ) y(x)^2+4 x^2 y(x)^2 y'(x)^2=0 \] ODE Classification
[[_homogeneous, `class G`], _rational, _dAlembert]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.797108 (sec), leaf count = 146
\[\left \{\{y(x)\to 0\},\text {Solve}\left [\log (y(x))=c_1+\frac {x^{5/2} \sqrt {4-\frac {x^3}{y(x)^2}} y(x) \sin ^{-1}\left (\frac {x^{3/2}}{2 y(x)}\right )}{\sqrt {x^8-4 x^5 y(x)^2}}+\frac {\log (x)}{2},y(x)\right ],\text {Solve}\left [\frac {x^{5/2} \sqrt {4-\frac {x^3}{y(x)^2}} y(x) \sin ^{-1}\left (\frac {x^{3/2}}{2 y(x)}\right )}{\sqrt {x^8-4 x^5 y(x)^2}}+\log (y(x))-\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.832 (sec), leaf count = 339
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}-{\frac {{x}^{3}}{4}}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{2\,{\it \_a}\, \left ( {{\it \_a}}^{3}-4\, \left ( y \left ( x \right ) \right ) ^{2} \right ) } \left ( {{\it \_a}}^{3}-4\, \left ( y \left ( x \right ) \right ) ^{2}-3\,\sqrt {{{\it \_a}}^{3} \left ( {{\it \_a}}^{3}-4\, \left ( y \left ( x \right ) \right ) ^{2} \right ) } \right ) }\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!4\,{\frac {{\it \_f}}{{x}^{3}-4\,{{\it \_f}}^{2}+\sqrt {{x}^{6}-4\,{{\it \_f}}^{2}{x}^{3}}}}-\int _{{\it \_b}}^{x}\!6\,{\frac {{\it \_f}}{{\it \_a}\, \left ( {{\it \_a}}^{3}-4\,{{\it \_f}}^{2} \right ) ^{2}} \left ( {\frac {{{\it \_a}}^{6}-4\,{{\it \_a}}^{3}{{\it \_f}}^{2}}{\sqrt {{{\it \_a}}^{3} \left ( {{\it \_a}}^{3}-4\,{{\it \_f}}^{2} \right ) }}}-2\,\sqrt {{{\it \_a}}^{3} \left ( {{\it \_a}}^{3}-4\,{{\it \_f}}^{2} \right ) } \right ) }\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{\frac {1}{2\,{\it \_a}\, \left ( {{\it \_a}}^{3}-4\, \left ( y \left ( x \right ) \right ) ^{2} \right ) } \left ( {{\it \_a}}^{3}-4\, \left ( y \left ( x \right ) \right ) ^{2}+3\,\sqrt {{{\it \_a}}^{3} \left ( {{\it \_a}}^{3}-4\, \left ( y \left ( x \right ) \right ) ^{2} \right ) } \right ) }\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-4\,{\frac {{\it \_f}}{-{x}^{3}+4\,{{\it \_f}}^{2}+\sqrt {{x}^{6}-4\,{{\it \_f}}^{2}{x}^{3}}}}-\int _{{\it \_b}}^{x}\!-6\,{\frac {{\it \_f}}{{\it \_a}\, \left ( {{\it \_a}}^{3}-4\,{{\it \_f}}^{2} \right ) ^{2}} \left ( {\frac {{{\it \_a}}^{6}-4\,{{\it \_a}}^{3}{{\it \_f}}^{2}}{\sqrt {{{\it \_a}}^{3} \left ( {{\it \_a}}^{3}-4\,{{\it \_f}}^{2} \right ) }}}-2\,\sqrt {{{\it \_a}}^{3} \left ( {{\it \_a}}^{3}-4\,{{\it \_f}}^{2} \right ) } \right ) }\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,y \left ( x \right ) =0 \right \} \] Mathematica raw input
DSolve[y[x]^2*(2*x^3 + y[x]^2) - 2*x*y[x]*(x^3 + 2*y[x]^2)*y'[x] + 4*x^2*y[x]^2*y'[x]^2 == 0,y[x],x]
Mathematica raw output
{{y[x] -> 0}, Solve[Log[y[x]] == C[1] + Log[x]/2 + (x^(5/2)*ArcSin[x^(3/2)/(2*y[x])]*Sqrt[4 - x^3/y[x]^2]*y[x])/Sqrt[x^8 - 4*x^5*y[x]^2], y[x]], Solve[-Log[x]/2 + Log[y[x]] + (x^(5/2)*ArcSin[x^(3/2)/(2*y[x])]*Sqrt[4 - x^3/y[x]^2]*y[x])/Sqrt[x^8 - 4*x^5*y[x]^2] == C[1], y[x]]}
Maple raw input
dsolve(4*x^2*y(x)^2*diff(y(x),x)^2-2*x*y(x)*(x^3+2*y(x)^2)*diff(y(x),x)+(2*x^3+y(x)^2)*y(x)^2 = 0, y(x),'implicit')
Maple raw output
y(x) = 0, y(x)^2-1/4*x^3 = 0, Int(1/2*(_a^3-4*y(x)^2+3*(_a^3*(_a^3-4*y(x)^2))^(1/2))/_a/(_a^3-4*y(x)^2),_a = _b .. x)+Intat(-4*_f/(-x^3+4*_f^2+(x^6-4*_f^2*x^3)^(1/2))-Int(-6*((_a^6-4*_a^3*_f^2)/(_a^3*(_a^3-4*_f^2))^(1/2)-2*(_a^3*(_a^3-4*_f^2))^(1/2))*_f/_a/(_a^3-4*_f^2)^2,_a = _b .. x),_f = y(x))+_C1 = 0, Int(1/2*(_a^3-4*y(x)^2-3*(_a^3*(_a^3-4*y(x)^2))^(1/2))/_a/(_a^3-4*y(x)^2),_a = _b .. x)+Intat(4*_f/(x^3-4*_f^2+(x^6-4*_f^2*x^3)^(1/2))-Int(6*((_a^6-4*_a^3*_f^2)/(_a^3*(_a^3-4*_f^2))^(1/2)-2*(_a^3*(_a^3-4*_f^2))^(1/2))*_f/_a/(_a^3-4*_f^2)^2,_a = _b .. x),_f = y(x))+_C1 = 0