Internal problem ID [9379]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1801 (book 6.210).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y \left (y^{2}+1\right ) y^{\prime \prime }+\left (1-3 y^{2}\right ) \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 75
dsolve(y(x)*(y(x)^2+1)*diff(diff(y(x),x),x)+(1-3*y(x)^2)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i \\ y \relax (x ) = i \\ y \relax (x ) = 0 \\ y \relax (x ) = -\frac {\sqrt {-2 \left (c_{1} x +c_{2}\right ) \left (2 c_{1} x +2 c_{2}+1\right )}}{2 \left (c_{1} x +c_{2}\right )} \\ y \relax (x ) = \frac {\sqrt {-2 \left (c_{1} x +c_{2}\right ) \left (2 c_{1} x +2 c_{2}+1\right )}}{2 c_{1} x +2 c_{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.358 (sec). Leaf size: 72
DSolve[(1 - 3*y[x]^2)*y'[x]^2 + y[x]*(1 + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {-1-2 c_1 (x+c_2)}}{\sqrt {2} \sqrt {c_1 (x+c_2)}} \\ y(x)\to \frac {\sqrt {-1-2 c_1 (x+c_2)}}{\sqrt {2} \sqrt {c_1 (x+c_2)}} \\ \end{align*}