Internal problem ID [9232]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1654 (book 6.63).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \left (\left (y^{\prime }\right )^{2}+1\right )^{\frac {3}{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 73
dsolve(diff(diff(y(x),x),x)=a*(diff(y(x),x)^2+1)^(3/2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = \frac {\left (a x +c_{1} a +1\right ) \left (a x +c_{1} a -1\right ) \sqrt {-\frac {1}{a^{2} x^{2}+2 a^{2} x c_{1}+a^{2} c_{1}^{2}-1}}}{a}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.543 (sec). Leaf size: 73
DSolve[-(a*(1 + y'[x]^2)^(3/2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {a c_2-i \sqrt {(a x-1+c_1) (a x+1+c_1)}}{a} \\ y(x)\to \frac {i \sqrt {(a x-1+c_1) (a x+1+c_1)}+a c_2}{a} \\ \end{align*}