Internal problem ID [9355]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1776.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) \left (y^{\prime }\right )^{2}-12 y^{\prime } y x^{2}+3 y^{2} x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 92
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), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {c_{2}^{2} \LegendreP \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right )^{2} x}{4 c_{1}}+c_{1} \LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right )^{2} x +c_{2} \LegendreP \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right ) x \LegendreQ \left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 54.706 (sec). Leaf size: 702
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,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \exp \left (\int _1^x-\frac {-2 \sqrt {K[2]^2+K[2]+1} \sqrt {\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}} \left (K[2] \left (4 K[2]+\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+4\right )+\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+4\right )+c_1 \sqrt {1-K[2]} \left (K[2]^2+K[2]+1\right ) \left (K[2] \left (4 K[2]+\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right )+3 \sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right )+\sqrt {1-K[2]} \left (K[2]^2+K[2]+1\right ) \left (K[2] \left (4 K[2]+\sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right )+3 \sqrt {2 K[2]+i \sqrt {3}+1} \sqrt {6 K[2]-3 i \sqrt {3}+3}+10\right ) \int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]}{2 (1-K[2])^{3/2} \sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1} \left (K[2]^2+K[2]+1\right ) \left (\sqrt {3} K[2]+\sqrt {2 K[2]-i \sqrt {3}+1} \sqrt {2 K[2]+i \sqrt {3}+1}+\sqrt {3}\right ) \left (c_1+\int _1^{K[2]}\frac {\sqrt {\sqrt {3} K[1]+\sqrt {2 K[1]-i \sqrt {3}+1} \sqrt {2 K[1]+i \sqrt {3}+1}+\sqrt {3}}}{2 (1-K[1])^{3/2} \sqrt {K[1]^2+K[1]+1}}dK[1]\right )}dK[2]\right ) \\ \end{align*}