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.
ODE order: 2.
ODE degree: 1.
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: 0.221 (sec). Leaf size: 36
dsolve(diff(diff(y(x),x),x)-a*(diff(y(x),x)^2+1)^(1/2)=0,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 (c_{1} a +a x \right )}{a}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 48.387 (sec). Leaf size: 50
DSolve[-(a*Sqrt[1 + y'[x]^2]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {1}{a \sqrt {\text {sech}^2(a x+c_1)}} \\ y(x)\to \frac {1}{a \sqrt {\text {sech}^2(a x+c_1)}}+c_2 \\ \end{align*}