Internal problem ID [9291]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1717 (book 6.126).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {y^{\prime \prime } y+a \left ({y^{\prime }}^{2}+1\right )=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 68
dsolve(diff(diff(y(x),x),x)*y(x)+a*(diff(y(x),x)^2+1)=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \left (x \right )}\frac {\textit {\_a}^{a}}{\sqrt {-\textit {\_a}^{2 a}+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \left (x \right )}-\frac {\textit {\_a}^{a}}{\sqrt {-\textit {\_a}^{2 a}+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.453 (sec). Leaf size: 172
DSolve[a*(1 + y'[x]^2) + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 c_1} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\text {$\#$1} \sqrt {1-e^{2 c_1} \text {$\#$1}^{-2 a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2 a},1-\frac {1}{2 a},e^{2 c_1} \text {$\#$1}^{-2 a}\right )}{\sqrt {-1+e^{2 c_1} \text {$\#$1}^{-2 a}}}\&\right ][x+c_2] \\ \end{align*}