23.4.61 problem 61

Internal problem ID [6363]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 61
Date solved : Tuesday, September 30, 2025 at 02:52:34 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.399 (sec). Leaf size: 34

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: 0.256 (sec). Leaf size: 19

ode=D[y[x],{x,2}] == a*Sqrt[1 + D[y[x],x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\cosh (a x+c_1)}{a}+c_2 \end{align*}

✓ Sympy. Time used: 155.273 (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)
 

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