ODE
\[ 8 \left (1-x^3\right ) y(x) y''(x)+4 \left (1-x^3\right ) y'(x)^2-12 x^2 y(x) y'(x)+3 x y(x)^2=0 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 1.50781 (sec), leaf count = 223
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x\frac {3 \left (5 \sqrt [3]{-1} \, _2F_1\left (\frac {1}{12} \left (17-\sqrt {10}\right ),\frac {1}{12} \left (17+\sqrt {10}\right );\frac {7}{3};K[1]^3\right ) K[1]-6 c_1 \, _2F_1\left (\frac {1}{12} \left (13-\sqrt {10}\right ),\frac {1}{12} \left (13+\sqrt {10}\right );\frac {5}{3};K[1]^3\right )\right ) K[1]^2+64 \sqrt [3]{-1} \, _2F_1\left (\frac {1}{12} \left (5-\sqrt {10}\right ),\frac {1}{12} \left (5+\sqrt {10}\right );\frac {4}{3};K[1]^3\right )}{96 \left (c_1 \, _2F_1\left (\frac {1}{12} \left (1-\sqrt {10}\right ),\frac {1}{12} \left (1+\sqrt {10}\right );\frac {2}{3};K[1]^3\right )+\sqrt [3]{-1} \, _2F_1\left (\frac {1}{12} \left (5-\sqrt {10}\right ),\frac {1}{12} \left (5+\sqrt {10}\right );\frac {4}{3};K[1]^3\right ) K[1]\right )}dK[1]\right )\right \}\right \}\]
Maple ✓
cpu = 1.29 (sec), leaf count = 438
\[\left [\textit {\_C2} \sqrt {x^{3}-1}\, \sqrt {-x^{3}+1}\, \sqrt {x}\, \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+\sqrt {x}\, \sqrt {x^{3}-1}\, \sqrt {-x^{3}+1}\, \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \textit {\_C1} -y \left (x \right )^{\frac {3}{2}} x^{3} \LegendreP \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \sqrt {10}\, \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+x^{3} \sqrt {10}\, y \left (x \right )^{\frac {3}{2}} \LegendreQ \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )-y \left (x \right )^{\frac {3}{2}} x^{3} \LegendreP \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+x^{3} y \left (x \right )^{\frac {3}{2}} \LegendreQ \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+y \left (x \right )^{\frac {3}{2}} \LegendreP \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \sqrt {10}\, \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )-\sqrt {10}\, y \left (x \right )^{\frac {3}{2}} \LegendreQ \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )+y \left (x \right )^{\frac {3}{2}} \LegendreP \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \LegendreQ \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right )-y \left (x \right )^{\frac {3}{2}} \LegendreQ \left (\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) \LegendreP \left (-\frac {1}{2}+\frac {\sqrt {10}}{6}, \frac {1}{3}, \sqrt {-x^{3}+1}\right ) = 0\right ]\] Mathematica raw input
DSolve[3*x*y[x]^2 - 12*x^2*y[x]*y'[x] + 4*(1 - x^3)*y'[x]^2 + 8*(1 - x^3)*y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^Inactive[Integrate][(64*(-1)^(1/3)*Hypergeometric2F1[(5 - Sqrt[10])/12, (5 + Sqrt[10])/12, 4/3, K[1]^3] + 3*K[1]^2*(-6*C[1]*Hypergeometric2F1[(13 - Sqrt[10])/12, (13 + Sqrt[10])/12, 5/3, K[1]^3] + 5*(-1)^(1/3)*Hypergeometric2F1[(17 - Sqrt[10])/12, (17 + Sqrt[10])/12, 7/3, K[1]^3]*K[1]))/(96*(C[1]*Hypergeometric2F1[(1 - Sqrt[10])/12, (1 + Sqrt[10])/12, 2/3, K[1]^3] + (-1)^(1/3)*Hypergeometric2F1[(5 - Sqrt[10])/12, (5 + Sqrt[10])/12, 4/3, K[1]^3]*K[1])), {K[1], 1, x}]*C[2]}}
Maple raw input
dsolve(8*(-x^3+1)*y(x)*diff(diff(y(x),x),x)+4*(-x^3+1)*diff(y(x),x)^2-12*x^2*y(x)*diff(y(x),x)+3*x*y(x)^2 = 0, y(x))
Maple raw output
[_C2*(x^3-1)^(1/2)*(-x^3+1)^(1/2)*x^(1/2)*LegendreQ(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))+x^(1/2)*(x^3-1)^(1/2)*(-x^3+1)^(1/2)*LegendreP(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*_C1-y(x)^(3/2)*x^3*LegendreP(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*10^(1/2)*LegendreQ(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))+x^3*10^(1/2)*y(x)^(3/2)*LegendreQ(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*LegendreP(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))-y(x)^(3/2)*x^3*LegendreP(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*LegendreQ(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))+x^3*y(x)^(3/2)*LegendreQ(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*LegendreP(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))+y(x)^(3/2)*LegendreP(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*10^(1/2)*LegendreQ(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))-10^(1/2)*y(x)^(3/2)*LegendreQ(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*LegendreP(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))+y(x)^(3/2)*LegendreP(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*LegendreQ(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))-y(x)^(3/2)*LegendreQ(1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2))*LegendreP(-1/2+1/6*10^(1/2),1/3,(-x^3+1)^(1/2)) = 0]