38.6.21 problem 21

Internal problem ID [8451]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 05:37:14 PM
CAS classification : [_linear]

\begin{align*} r^{\prime }+r \sec \left (t \right )&=\cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

ode:=diff(r(t),t)+r(t)*sec(t) = cos(t); 
dsolve(ode,r(t), singsol=all);
 

\[ r = \frac {\left (t -\cos \left (t \right )+c_1 \right ) \left (\cos \left (t \right )-\sin \left (t \right )+1\right )}{\cos \left (t \right )+\sin \left (t \right )+1} \]

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 43

ode=D[r[t],t]+r[t]*Sec[t]==Cos[t]; 
ic={}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\begin{align*} r(t)&\to e^{-2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \left (\int _1^te^{2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right )} \cos (K[1])dK[1]+c_1\right ) \end{align*}

✓ Sympy. Time used: 14.702 (sec). Leaf size: 119

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

\[ r{\left (t \right )} = \frac {\sqrt {\sin {\left (t \right )} - 1} \left (C_{1} - \sqrt {\sin {\left (t \right )} - 1} \sqrt {\sin {\left (t \right )} + 1} - 2 \log {\left (2 \sqrt {\sin {\left (t \right )} - 1} + 2 \sqrt {\sin {\left (t \right )} + 1} \right )} - \int \left (- \frac {\sqrt {\sin {\left (t \right )} + 1} r{\left (t \right )}}{\sqrt {\sin {\left (t \right )} - 1} \cos {\left (t \right )}}\right )\, dt\right )}{\sqrt {\sin {\left (t \right )} - 1} \int \frac {\sqrt {\sin {\left (t \right )} + 1}}{\sqrt {\sin {\left (t \right )} - 1} \cos {\left (t \right )}}\, dt - \sqrt {\sin {\left (t \right )} + 1}} \]