69.14.15 problem 341

Internal problem ID [18216]
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 : Thursday, October 02, 2025 at 03:09:14 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.435 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x) = (1+diff(y(x),x)^2)^(1/2); 
dsolve(ode,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*}

✓ Mathematica. Time used: 4.324 (sec). Leaf size: 50

ode=D[y[x],{x,2}]==Sqrt[1+D[y[x],x]^2]; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(Derivative(y(x), x)**2 + 1) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out