65.2.3 problem 7.1 (iii)

Internal problem ID [13637]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 7, Scalar autonomous ODEs. Exercises page 56
Problem number : 7.1 (iii)
Date solved : Wednesday, March 05, 2025 at 10:05:49 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

\[ t +\int _{}^{x \left (t \right )}\frac {\csc \left (\textit {\_a} \right )}{\left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} +c_{1} = 0 \]

✓ Mathematica. Time used: 0.265 (sec). Leaf size: 52

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

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\csc (K[1])}{(K[1]-2) (K[1]+1)}dK[1]\&\right ][-t+c_1] \\ x(t)\to -1 \\ x(t)\to 0 \\ x(t)\to 2 \\ \end{align*}

✓ Sympy. Time used: 0.876 (sec). Leaf size: 17

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

\[ \int \limits ^{x{\left (t \right )}} \frac {1}{\left (y - 2\right ) \left (y + 1\right ) \sin {\left (y \right )}}\, dy = C_{1} - t \]