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]
Book solution method
Change of variable
Mathematica ✓
cpu = 1.76837 (sec), leaf count = 146
\[\left \{\{y(x)\to 0\},\text {Solve}\left [\log (y(x))=\frac {x^{5/2} \sqrt {4-\frac {x^3}{y(x)^2}} y(x) \sin ^{-1}\left (\frac {x^{3/2}}{2 y(x)}\right )}{\sqrt {x^8-4 x^5 y(x)^2}}+\frac {\log (x)}{2}+c_1,y(x)\right ],\text {Solve}\left [\frac {x^{5/2} \sqrt {4-\frac {x^3}{y(x)^2}} y(x) \sin ^{-1}\left (\frac {x^{3/2}}{2 y(x)}\right )}{\sqrt {x^8-4 x^5 y(x)^2}}+\log (y(x))-\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.594 (sec), leaf count = 391
\[\left [y \left (x \right ) = 0, y \left (x \right ) = -\frac {x^{\frac {3}{2}}}{2}, y \left (x \right ) = \frac {x^{\frac {3}{2}}}{2}, \int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{3}-4 y \left (x \right )^{2}+3 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4 y \left (x \right )^{2}\right )}}{2 \textit {\_a} \left (\textit {\_a}^{3}-4 y \left (x \right )^{2}\right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (-\frac {4 \textit {\_f}}{-x^{3}+4 \textit {\_f}^{2}+\sqrt {x^{6}-4 \textit {\_f}^{2} x^{3}}}-\left (\int _{\textit {\_b}}^{x}\left (\frac {-8 \textit {\_f} -\frac {12 \textit {\_a}^{3} \textit {\_f}}{\sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )}}}{2 \textit {\_a} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )}+\frac {4 \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}+3 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )}\right ) \textit {\_f}}{\textit {\_a} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +\textit {\_C1} = 0, \int _{\textit {\_b}}^{x}-\frac {-\textit {\_a}^{3}+4 y \left (x \right )^{2}+3 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4 y \left (x \right )^{2}\right )}}{2 \textit {\_a} \left (\textit {\_a}^{3}-4 y \left (x \right )^{2}\right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (\frac {4 \textit {\_f}}{x^{3}-4 \textit {\_f}^{2}+\sqrt {x^{6}-4 \textit {\_f}^{2} x^{3}}}-\left (\int _{\textit {\_b}}^{x}\left (-\frac {8 \textit {\_f} -\frac {12 \textit {\_a}^{3} \textit {\_f}}{\sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )}}}{2 \textit {\_a} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )}-\frac {4 \left (-\textit {\_a}^{3}+4 \textit {\_f}^{2}+3 \sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )}\right ) \textit {\_f}}{\textit {\_a} \left (\textit {\_a}^{3}-4 \textit {\_f}^{2}\right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +\textit {\_C1} = 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[-1/2*Log[x] + 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))
Maple raw output
[y(x) = 0, y(x) = -1/2*x^(3/2), y(x) = 1/2*x^(3/2), 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(1/2*(-8*_f-12/(_a^3*(_a^3-4*_f^2))^(1/2)*_a^3*_f)/_a/(_a^3-4*_f^2)+4*(_a^3-4*_f^2+3*(_a^3*(_a^3-4*_f^2))^(1/2))/_a/(_a^3-4*_f^2)^2*_f,_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(-1/2*(8*_f-12/(_a^3*(_a^3-4*_f^2))^(1/2)*_a^3*_f)/_a/(_a^3-4*_f^2)-4*(-_a^3+4*_f^2+3*(_a^3*(_a^3-4*_f^2))^(1/2))/_a/(_a^3-4*_f^2)^2*_f,_a = _b .. x),_f = y(x))+_C1 = 0]