Internal problem ID [9229]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1650 (book 6.60).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a \sqrt {\left (y^{\prime }\right )^{2}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.36 (sec). Leaf size: 36
dsolve(diff(diff(y(x),x),x)=a*(diff(y(x),x)^2+1)^(1/2),y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = \frac {\cosh \left (a x +c_{1} a \right )}{a}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.463 (sec). Leaf size: 19
DSolve[-(a*Sqrt[1 + y'[x]^2]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\cosh (a x+c_1)}{a}+c_2 \\ \end{align*}