54.7.47 problem 1650 (book 6.60)

Internal problem ID [12896]
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 : Wednesday, October 01, 2025 at 02:45:06 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.388 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x) = a*(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 &= c_{2} +\frac {\cosh \left (a \left (c_{1} +x \right )\right )}{a} \\ \end{align*}

✓ Mathematica. Time used: 6.637 (sec). Leaf size: 44

ode=-(a*Sqrt[1 + D[y[x],x]^2]) + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 160.431 (sec). Leaf size: 48

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {\cosh {\left (C_{2} - a x \right )}}{a}, \ y{\left (x \right )} = C_{1} + \frac {\cosh {\left (C_{2} + a x \right )}}{a}, \ y{\left (x \right )} = C_{1} - \frac {\cosh {\left (C_{2} - a x \right )}}{a}, \ y{\left (x \right )} = C_{1} + \frac {\cosh {\left (C_{2} + a x \right )}}{a}\right ] \]