86.9.2 problem 10

Internal problem ID [23172]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 7. Polar coordinates and vectors. Exercise 7a at page 109
Problem number : 10
Date solved : Thursday, October 02, 2025 at 09:23:46 PM
CAS classification : [_separable]

\begin{align*} \frac {r^{\prime }}{r}&=\tan \left (\theta \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 8

ode:=1/r(theta)*diff(r(theta),theta) = tan(theta); 
dsolve(ode,r(theta), singsol=all);
 

\[ r = c_1 \sec \left (\theta \right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 10

ode=1/r[\[Theta]]*D[r[\[Theta]],\[Theta]]==Tan[\[Theta]]; 
ic={}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to c_1 \sec (\theta ) \end{align*}

✓ Sympy. Time used: 0.177 (sec). Leaf size: 7

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

\[ r{\left (t \right )} = \frac {C_{1}}{\cos {\left (t \right )}} \]