80.2.13 problem 19

Internal problem ID [21156]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 2. Theory of first order differential equations. Excercise 2.6 at page 37
Problem number : 19
Date solved : Thursday, October 02, 2025 at 07:11:14 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.084 (sec). Leaf size: 29

ode:=diff(x(t),t) = 2+sin(x(t)); 
ic:=[x(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = 2 \arctan \left (\frac {\sqrt {3}\, \tan \left (\frac {\left (\sqrt {3}\, \pi +9 t \right ) \sqrt {3}}{18}\right )}{2}-\frac {1}{2}\right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 70

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

\begin{align*} x(t)&\to -2 \arctan \left (\frac {1}{2} \left (1-\sqrt {3} \tan \left (\frac {1}{6} \left (3 \sqrt {3} t+\pi \right )\right )\right )\right )\\ x(t)&\to 2 \arctan \left (\frac {1}{2} \left (\sqrt {3} \tan \left (\frac {1}{6} \left (3 \sqrt {3} t+\pi \right )\right )-1\right )\right ) \end{align*}

✗ Sympy

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

Timed Out