70.1.18 problem 2.4 (v)

Internal problem ID [14237]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.4 (v)
Date solved : Monday, March 31, 2025 at 12:13:44 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }&=\left (2 \cos \left (x\right )-1\right ) \sin \left (x\right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

ode:=diff(diff(x(t),t),t) = (2*cos(x(t))-1)*sin(x(t)); 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 48

ode=D[x[t],{t,2}]==(2*Cos[x[t]]-1)*Sin[x[t]]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{x(t)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}(\sin (2 K[1])-\sin (K[1]))dK[1]}}dK[2]{}^2=(t+c_2){}^2,x(t)\right ] \]

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((1 - 2*cos(x(t)))*sin(x(t)) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out