82.8.33 problem 36-33

Internal problem ID [21904]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-33
Date solved : Thursday, October 02, 2025 at 08:05:54 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 2.255 (sec). Leaf size: 69

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

\begin{align*} x(t)&\to -2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {(c_1+2) (t+c_2){}^2},\frac {4}{c_1+2}\right )\\ x(t)&\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {(c_1+2) (t+c_2){}^2},\frac {4}{c_1+2}\right ) \end{align*}

✗ Sympy

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

Timed Out