Internal problem ID [9361]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1782.
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 {\left (y^{2}+1\right ) y^{\prime \prime }-3 \left (y^{\prime }\right )^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.119 (sec). Leaf size: 66
dsolve((y(x)^2+1)*diff(diff(y(x),x),x)-3*y(x)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i \\ y \relax (x ) = i \\ y \relax (x ) = c_{1} x \sqrt {-\frac {1}{c_{1}^{2} x^{2}+2 c_{1} c_{2} x +c_{2}^{2}-1}}+c_{2} \sqrt {-\frac {1}{c_{1}^{2} x^{2}+2 c_{1} c_{2} x +c_{2}^{2}-1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.134 (sec). Leaf size: 63
DSolve[-3*y[x]*y'[x]^2 + (1 + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i c_1 (x+c_2)}{\sqrt {-1+c_1{}^2 (x+c_2){}^2}} \\ y(x)\to \frac {i c_1 (x+c_2)}{\sqrt {-1+c_1{}^2 (x+c_2){}^2}} \\ \end{align*}