Internal problem ID [9179]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1600.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+d +y^{2} b +c y+a y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 89
dsolve(diff(diff(y(x),x),x)+d+y(x)^2*b+c*y(x)+a*y(x)^3=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}-\frac {6}{\sqrt {-18 a \,\textit {\_a}^{4}-24 b \,\textit {\_a}^{3}-36 c \,\textit {\_a}^{2}-72 d \textit {\_a} +36 c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {6}{\sqrt {-18 a \,\textit {\_a}^{4}-24 b \,\textit {\_a}^{3}-36 c \,\textit {\_a}^{2}-72 d \textit {\_a} +36 c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.213 (sec). Leaf size: 79
DSolve[1 - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} e^{-e^{c_1} (x+c_2)-c_1} \left (-1+e^{2 e^{c_1} (x+c_2)}\right ) \\ y(x)\to \frac {1}{2} e^{-e^{c_1} (x+c_2)-c_1} \left (-1+e^{2 e^{c_1} (x+c_2)}\right ) \\ \end{align*}