Internal problem ID [9360]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1781.
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 }+\left (1-2 y\right ) \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.123 (sec). Leaf size: 21
dsolve((y(x)^2+1)*diff(diff(y(x),x),x)+(1-2*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 ) = \tan \left (\ln \left (x c_{1}+c_{2}\right )\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.164 (sec). Leaf size: 45
DSolve[(1 - 2*y[x])*y'[x]^2 + (1 + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (-1+c_1{}^{2 i} (x+c_2){}^{2 i}\right )}{1+c_1{}^{2 i} (x+c_2){}^{2 i}} \\ \end{align*}