72.2.26 problem 19 a(iii)

Internal problem ID [14582]
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 1.3 page 47
Problem number : 19 a(iii)
Date solved : Thursday, March 13, 2025 at 03:37:11 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.203 (sec). Leaf size: 50

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

\begin{align*} \theta (t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{9 \cos (K[1])-11}dK[1]\&\right ]\left [-\frac {t}{10}+c_1\right ] \\ \theta (t)\to -\arccos \left (\frac {11}{9}\right ) \\ \theta (t)\to \arccos \left (\frac {11}{9}\right ) \\ \end{align*}

✓ Sympy. Time used: 1.232 (sec). Leaf size: 34

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

\[ - \sqrt {10} \left (\operatorname {atan}{\left (\sqrt {10} \tan {\left (\frac {\theta {\left (t \right )}}{2} \right )} \right )} + \pi \left \lfloor {\frac {\theta {\left (t \right )} - \pi }{2 \pi }}\right \rfloor \right ) = C_{1} - t \]