Internal problem ID [9973]
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]]
\[ \boxed {y^{\prime \prime }-a \sqrt {{y^{\prime }}^{2}+1}=0} \]
✓ Solution by Maple
Time used: 0.219 (sec). Leaf size: 34
dsolve(diff(diff(y(x),x),x)=a*(diff(y(x),x)^2+1)^(1/2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= c_{2} +\frac {\cosh \left (a \left (c_{1} +x \right )\right )}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.728 (sec). Leaf size: 35
DSolve[-(a*Sqrt[1 + y'[x]^2]) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{a x+c_1} \left (1+e^{-2 (a x+c_1)}\right )}{2 a}+c_2 \]