50.16.3 problem 4

Internal problem ID [8071]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Problems for Discussion and Exploration. Page 105
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:42:07 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 3.971 (sec). Leaf size: 69

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

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

✗ Sympy

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

Timed Out