ODE
\[ 2 y(x)^3 y'(x)=x^3-x y(x)^2 \] ODE Classification
[[_homogeneous, `class A`], _rational, _dAlembert]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.492732 (sec), leaf count = 714
\[\left \{\left \{y(x)\to -\frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-x^2+\frac {x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}}{\sqrt {2}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}-2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x^4}{\sqrt [3]{x^6+2 \sqrt {e^{24 c_1}-e^{12 c_1} x^6}-2 e^{12 c_1}}}}\right \}\right \}\]
Maple ✓
cpu = 0.514 (sec), leaf count = 711
\[\left [y \left (x \right ) = -\frac {\sqrt {2 \left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}+\frac {2 x^{4} \textit {\_C1}^{2}}{\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}-2 x^{2} \textit {\_C1}}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {\sqrt {2 \left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}+\frac {2 x^{4} \textit {\_C1}^{2}}{\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}-2 x^{2} \textit {\_C1}}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = -\frac {\sqrt {-\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}-\frac {x^{4} \textit {\_C1}^{2}}{\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}-2 x^{2} \textit {\_C1} -2 i \sqrt {3}\, \left (\frac {\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{2}-\frac {x^{4} \textit {\_C1}^{2}}{2 \left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}\right )}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {\sqrt {-\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}-\frac {x^{4} \textit {\_C1}^{2}}{\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}-2 x^{2} \textit {\_C1} -2 i \sqrt {3}\, \left (\frac {\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{2}-\frac {x^{4} \textit {\_C1}^{2}}{2 \left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}\right )}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = -\frac {\sqrt {-\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}-\frac {x^{4} \textit {\_C1}^{2}}{\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}-2 x^{2} \textit {\_C1} +2 i \sqrt {3}\, \left (\frac {\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{2}-\frac {x^{4} \textit {\_C1}^{2}}{2 \left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}\right )}}{2 \sqrt {\textit {\_C1}}}, y \left (x \right ) = \frac {\sqrt {-\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}-\frac {x^{4} \textit {\_C1}^{2}}{\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}-2 x^{2} \textit {\_C1} +2 i \sqrt {3}\, \left (\frac {\left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}{2}-\frac {x^{4} \textit {\_C1}^{2}}{2 \left (2+\textit {\_C1}^{3} x^{6}+2 \sqrt {\textit {\_C1}^{3} x^{6}+1}\right )^{\frac {1}{3}}}\right )}}{2 \sqrt {\textit {\_C1}}}\right ]\] Mathematica raw input
DSolve[2*y[x]^3*y'[x] == x^3 - x*y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[-x^2 + x^4/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/Sqrt[2])}, {y[x] -> Sqrt[-x^2 + x^4/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/Sqrt[2]}, {y[x] -> -1/2*Sqrt[-2*x^2 + (I*(I + Sqrt[3])*x^4)/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-1 - I*Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]}, {y[x] -> Sqrt[-2*x^2 + (I*(I + Sqrt[3])*x^4)/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-1 - I*Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/2}, {y[x] -> -1/2*Sqrt[-2*x^2 + ((-1 - I*Sqrt[3])*x^4)/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + I*(I + Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]}, {y[x] -> Sqrt[-2*x^2 + ((-1 - I*Sqrt[3])*x^4)/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + I*(I + Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/2}}
Maple raw input
dsolve(2*y(x)^3*diff(y(x),x) = x^3-x*y(x)^2, y(x))
Maple raw output
[y(x) = -1/2*(2*(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)+2*x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-2*x^2*_C1)^(1/2)/_C1^(1/2), y(x) = 1/2*(2*(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)+2*x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-2*x^2*_C1)^(1/2)/_C1^(1/2), y(x) = -1/2*(-(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-2*x^2*_C1-2*I*3^(1/2)*(1/2*(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-1/2*x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)))^(1/2)/_C1^(1/2), y(x) = 1/2*(-(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-2*x^2*_C1-2*I*3^(1/2)*(1/2*(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-1/2*x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)))^(1/2)/_C1^(1/2), y(x) = -1/2*(-(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-2*x^2*_C1+2*I*3^(1/2)*(1/2*(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-1/2*x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)))^(1/2)/_C1^(1/2), y(x) = 1/2*(-(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-2*x^2*_C1+2*I*3^(1/2)*(1/2*(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)-1/2*x^4*_C1^2/(2+_C1^3*x^6+2*(_C1^3*x^6+1)^(1/2))^(1/3)))^(1/2)/_C1^(1/2)]