82.51.1 problem Ex. 1

Internal problem ID [18929]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at page 114
Problem number : Ex. 1
Date solved : Thursday, March 13, 2025 at 01:12:14 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.572 (sec). Leaf size: 52

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

\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {4 c_{2}^{2} {\mathrm e}^{\frac {x}{a}}+a^{2} {\mathrm e}^{-\frac {x}{a}}+4 c_{2} c_{1}}{4 c_{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.562 (sec). Leaf size: 40

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

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

✗ Sympy

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

Timed Out