75.14.15 problem 341

Internal problem ID [16929]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 341
Date solved : Tuesday, January 28, 2025 at 09:41:09 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Solution by Maple

Time used: 1.450 (sec). Leaf size: 26

dsolve(diff(y(x),x$2)=sqrt(1+diff(y(x),x)^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 8.071 (sec). Leaf size: 50

DSolve[D[y[x],{x,2}]==Sqrt[1+D[y[x],x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \int _1^x\sinh (c_1+K[1])dK[1]+c_2 \\ y(x)\to \frac {1}{2} \left (e^{-x}+e^x-e-\frac {1}{e}+2 c_2\right ) \\ \end{align*}