Internal problem ID [9351]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1777 (book 6.186).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {8 \left (-x^{3}+1\right ) y y^{\prime \prime }-4 \left (-x^{3}+1\right ) {y^{\prime }}^{2}-12 y y^{\prime } x^{2}+3 y^{2} x=0} \]
✓ Solution by Maple
Time used: 0.14 (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 \left (x \right ) = 0 \\ y \left (x \right ) = \frac {c_{2}^{2} {\operatorname {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} {\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right )}^{2} x +c_{2} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \sqrt {-\left (x -1\right ) \left (x^{2}+x +1\right )}\right ) x \operatorname {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: 0.0 (sec). Leaf size: 0
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]
Timed out