87.29.16 problem 16

Internal problem ID [23896]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 304
Problem number : 16
Date solved : Sunday, October 12, 2025 at 05:55:16 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=\sin \left (x_{1} \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.247 (sec). Leaf size: 77

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = sin(x__1(t))]; 
dsolve(ode);
 

\begin{align*} \left \{x_{2} \left (t \right ) &= \operatorname {RootOf}\left (-2 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {-\textit {\_f}^{4}-4 c_1 \,\textit {\_f}^{2}-4 c_1^{2}+4}}d \textit {\_f} +t +c_2 \right ), x_{2} \left (t \right ) = \operatorname {RootOf}\left (2 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {-\textit {\_f}^{4}-4 c_1 \,\textit {\_f}^{2}-4 c_1^{2}+4}}d \textit {\_f} +t +c_2 \right )\right \} \\ \{x_{1} \left (t \right ) &= \arcsin \left (\frac {d}{d t}x_{2} \left (t \right )\right )\} \\ \end{align*}

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 182

ode={D[x1[t],t]==x2[t],D[x2[t],t]==Sin[x1[t]]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x2}(t)&\to -\sqrt {2} \sqrt {c_1-\cos \left (2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1-1} \left (c_2-\sqrt {2} t\right ),-\frac {2}{c_1-1}\right )\right )}\\ \text {x1}(t)&\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1-1} \left (c_2-\sqrt {2} t\right ),-\frac {2}{c_1-1}\right )\\ \text {x2}(t)&\to \sqrt {2} \sqrt {c_1-\cos \left (2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1-1} \left (\sqrt {2} t+c_2\right ),-\frac {2}{c_1-1}\right )\right )}\\ \text {x1}(t)&\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1-1} \left (\sqrt {2} t+c_2\right ),-\frac {2}{c_1-1}\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(-sin(x1(t)) + Derivative(x2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)