Internal problem ID [9296]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1717.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime } y+a \left (y^{\prime }\right )^{2}+b y^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 112
dsolve(diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)^3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \int _{}^{y \relax (x )}\frac {\left (2 a +3\right ) \textit {\_a}^{2 a}}{\sqrt {-\left (2 a +3\right ) \textit {\_a}^{2 a} \left (2 b \,\textit {\_a}^{2 a +3}-c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\left (2 a +3\right ) \textit {\_a}^{2 a}}{\sqrt {-\left (2 a +3\right ) \textit {\_a}^{2 a} \left (2 b \,\textit {\_a}^{2 a +3}-c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 47.664 (sec). Leaf size: 277
DSolve[b*y[x]^3 + a*y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {y(x) \sqrt {(2 a+3) y(x)^{2 a}} \sqrt {1-\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}} \, _2F_1\left (\frac {1}{2},\frac {a+1}{2 a+3};\frac {a+1}{2 a+3}+1;\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}\right )}{(a+1) \sqrt {-2 b y(x)^{2 a+3}+2 a c_1+3 c_1}}=-x+c_2,y(x)\right ] \\ \text {Solve}\left [\frac {y(x) \sqrt {(2 a+3) y(x)^{2 a}} \sqrt {1-\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}} \, _2F_1\left (\frac {1}{2},\frac {a+1}{2 a+3};\frac {a+1}{2 a+3}+1;\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}\right )}{(a+1) \sqrt {-2 b y(x)^{2 a+3}+2 a c_1+3 c_1}}=x+c_2,y(x)\right ] \\ \end{align*}