80.8.7 problem 11

Internal problem ID [21364]
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 8. Qualitative analysis of 2 by 2 systems and nonlinear second order equations. Excercise 8.5 at page 184
Problem number : 11
Date solved : Thursday, October 02, 2025 at 07:28:48 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} L x^{\prime \prime }+g \sin \left (x\right )&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 60

ode:=L*diff(diff(x(t),t),t)+g*sin(x(t)) = 0; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} L \int _{}^{x}\frac {1}{\sqrt {L \left (c_1 L +2 g \cos \left (\textit {\_a} \right )\right )}}d \textit {\_a} -t -c_2 &= 0 \\ -L \int _{}^{x}\frac {1}{\sqrt {L \left (c_1 L +2 g \cos \left (\textit {\_a} \right )\right )}}d \textit {\_a} -t -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 3.312 (sec). Leaf size: 91

ode=L*D[x[t],{t,2}]+g*Sin[x[t]] ==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 2 \operatorname {JacobiAmplitude}\left (\frac {\sqrt {2 g+L c_1} (t+c_2)}{2 \sqrt {L}},\frac {4 g}{2 g+L c_1}\right )\\ x(t)&\to -2 \operatorname {JacobiAmplitude}\left (\frac {\sqrt {2 g+L c_1} (t+c_2)}{2 \sqrt {L}},\frac {4 g}{2 g+L c_1}\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
L = symbols("L") 
g = symbols("g") 
x = Function("x") 
ode = Eq(L*Derivative(x(t), (t, 2)) + g*sin(x(t)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out