60.7.60 problem 1650 (book 6.60)

Internal problem ID [11649]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1650 (book 6.60)
Date solved : Monday, January 27, 2025 at 11:29:05 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=a \sqrt {{y^{\prime }}^{2}+1} \end{align*}

✓ Solution by Maple

Time used: 0.902 (sec). Leaf size: 32

dsolve(diff(diff(y(x),x),x)=a*(diff(y(x),x)^2+1)^(1/2),y(x), singsol=all)
 

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= c_{2} +\frac {\cosh \left (\left (x +c_{1} \right ) a \right )}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 8.050 (sec). Leaf size: 44

DSolve[-(a*Sqrt[1 + D[y[x],x]^2]) + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \int _1^x\sinh (c_1+a K[1])dK[1]+c_2 \\ y(x)\to \frac {\cosh (a x)-\cosh (a)+a c_2}{a} \\ \end{align*}