Internal problem ID [9225]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1646.
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 }+a y \left (\left (y^{\prime }\right )^{2}+1\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 94
dsolve(diff(diff(y(x),x),x)+a*y(x)*(diff(y(x),x)^2+1)^2=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {a \left (\textit {\_a}^{2}+2 c_{1}\right )}{\sqrt {-a \left (\textit {\_a}^{2}+2 c_{1}\right ) \left (\textit {\_a}^{2} a +2 c_{1} a -1\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {a \left (\textit {\_a}^{2}+2 c_{1}\right )}{\sqrt {-a \left (\textit {\_a}^{2}+2 c_{1}\right ) \left (\textit {\_a}^{2} a +2 c_{1} a -1\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.489 (sec). Leaf size: 262
DSolve[a*y[x]*(1 + y'[x]^2)^2 + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 (-a)+1+2 c_1}{1+2 c_1}} \sqrt {2 \text {$\#$1}^2 a-4 c_1} E\left (\text {ArcSin}\left (\sqrt {\frac {a}{2 c_1+1}} \text {$\#$1}\right )|1+\frac {1}{2 c_1}\right )}{\sqrt {\frac {a}{1+2 c_1}} \sqrt {\text {$\#$1}^2 (-a)+1+2 c_1} \sqrt {2-\frac {\text {$\#$1}^2 a}{c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 (-a)+1+2 c_1}{1+2 c_1}} \sqrt {2 \text {$\#$1}^2 a-4 c_1} E\left (\text {ArcSin}\left (\sqrt {\frac {a}{2 c_1+1}} \text {$\#$1}\right )|1+\frac {1}{2 c_1}\right )}{\sqrt {\frac {a}{1+2 c_1}} \sqrt {\text {$\#$1}^2 (-a)+1+2 c_1} \sqrt {2-\frac {\text {$\#$1}^2 a}{c_1}}}\&\right ][x+c_2] \\ \end{align*}