80.12.10 problem 17

Internal problem ID [21443]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 13. Boundary value problems. Excercise 13.5 at page 291
Problem number : 17
Date solved : Thursday, October 02, 2025 at 07:32:21 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} -x^{\prime \prime }&=\frac {1}{\sqrt {x^{2}+1}}-x \end{align*}

With initial conditions

\begin{align*} x \left (a \right )&=0 \\ x \left (b \right )&=0 \\ \end{align*}

✗ Maple

ode:=-diff(diff(x(t),t),t) = 1/(1+x(t)^2)^(1/2)-x(t); 
ic:=[x(a) = 0, x(b) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=-D[x[t],{t,2}]==1/Sqrt[1+x[t]^2]-x[t]; 
ic={x[a]==0,x[b]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

{}

✗ Sympy

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

Timed Out