72.3.13 problem 17

Internal problem ID [14606]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.4 page 61
Problem number : 17
Date solved : Monday, March 31, 2025 at 12:40:17 PM
CAS classification : [_quadrature]

\begin{align*} \theta ^{\prime }&=\frac {9}{10}-\frac {11 \cos \left (\theta \right )}{10} \end{align*}

With initial conditions

\begin{align*} \theta \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.249 (sec). Leaf size: 29

ode:=diff(theta(t),t) = 9/10-11/10*cos(theta(t)); 
ic:=theta(0) = 1; 
dsolve([ode,ic],theta(t), singsol=all);
 

\[ \theta = -2 \arctan \left (\frac {\tanh \left (-\operatorname {arctanh}\left (\tan \left (\frac {1}{2}\right ) \sqrt {10}\right )+\frac {\sqrt {10}\, t}{10}\right ) \sqrt {10}}{10}\right ) \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 36

ode=D[ theta[t],t]==1-Cos[ theta[t]] + (1+Cos[theta[t]])*(-1/10); 
ic={theta[0]==1}; 
DSolve[{ode,ic},theta[t],t,IncludeSingularSolutions->True]
 

\[ \theta (t)\to -2 \arctan \left (\frac {\tanh \left (\frac {t}{\sqrt {10}}-\text {arctanh}\left (\sqrt {10} \tan \left (\frac {1}{2}\right )\right )\right )}{\sqrt {10}}\right ) \]

✓ Sympy. Time used: 4.109 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
theta = Function("theta") 
ode = Eq(11*cos(theta(t))/10 + Derivative(theta(t), t) - 9/10,0) 
ics = {theta(0): 1} 
dsolve(ode,func=theta(t),ics=ics)
 

\[ t - \frac {\sqrt {10} \log {\left (\tan {\left (\frac {\theta {\left (t \right )}}{2} \right )} - \frac {\sqrt {10}}{10} \right )}}{2} + \frac {\sqrt {10} \log {\left (\tan {\left (\frac {\theta {\left (t \right )}}{2} \right )} + \frac {\sqrt {10}}{10} \right )}}{2} = \frac {\sqrt {10} \log {\left (\frac {\sqrt {10}}{10} + \tan {\left (\frac {1}{2} \right )} \right )}}{2} - \frac {\sqrt {10} \log {\left (- \frac {\sqrt {10}}{10} + \tan {\left (\frac {1}{2} \right )} \right )}}{2} \]