29.7.8 problem 5

Internal problem ID [7325]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:28:54 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} {y^{\prime \prime }}^{2}&=k^{2} \left (1+{y^{\prime }}^{2}\right ) \end{align*}

✓ Maple. Time used: 0.685 (sec). Leaf size: 65

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

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= c_{1} +c_{2} {\mathrm e}^{k x}+\frac {{\mathrm e}^{-k x}}{4 k^{2} c_{2}} \\ y &= c_{1} +\frac {{\mathrm e}^{k x}}{4 c_{2} k^{2}}+c_{2} {\mathrm e}^{-k x} \\ \end{align*}

✓ Mathematica. Time used: 11.032 (sec). Leaf size: 88

ode=(D[y[x],{x,2}])^2==k^2*(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 K[2])dK[2]+c_2\\ y(x)&\to \int _1^x\sinh (c_1+k K[3])dK[3]+c_2\\ y(x)&\to \frac {-\cosh (k x)+\cosh (k)+c_2 k}{k}\\ y(x)&\to \frac {\cosh (k x)-\cosh (k)+c_2 k}{k} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - sqrt((-k + Derivative(y(x), (x, 2)))*(k + Derivative(y(x), (x, 2))))/k cannot be solved by the factorable group method