87.29.13 problem 13

Internal problem ID [23893]
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 : 13
Date solved : Sunday, October 12, 2025 at 05:55:16 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 \sin \left (x \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.261 (sec). Leaf size: 81

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -4*sin(x(t))]; 
dsolve(ode);
 

\begin{align*} \left \{y \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}+64}}d \textit {\_f} +t +c_2 \right ), y \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}+64}}d \textit {\_f} +t +c_2 \right )\right \} \\ \left \{x \left (t \right ) &= -\arcsin \left (\frac {\frac {d}{d t}y \left (t \right )}{4}\right )\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 182

ode={D[x[t],t]==y[t],D[y[t],t]==-4*Sin[x[t]]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sqrt {2} \sqrt {4 \cos \left (2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1+4} \left (c_2-\sqrt {2} t\right ),\frac {8}{c_1+4}\right )\right )+c_1}\\ x(t)&\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1+4} \left (c_2-\sqrt {2} t\right ),\frac {8}{c_1+4}\right )\\ y(t)&\to \sqrt {2} \sqrt {4 \cos \left (2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1+4} \left (\sqrt {2} t+c_2\right ),\frac {8}{c_1+4}\right )\right )+c_1}\\ x(t)&\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {c_1+4} \left (\sqrt {2} t+c_2\right ),\frac {8}{c_1+4}\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(4*sin(x(t)) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)