86.9.7 problem 15

Internal problem ID [23177]
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 : 15
Date solved : Thursday, October 02, 2025 at 09:23:59 PM
CAS classification : [_quadrature]

\begin{align*} \sin \left (\theta \right )^{2} r^{\prime }&=-b \cos \left (\theta \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 10

ode:=sin(theta)^2*diff(r(theta),theta) = -b*cos(theta); 
dsolve(ode,r(theta), singsol=all);
 

\[ r = b \csc \left (\theta \right )+c_1 \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 12

ode=Sin[\[Theta]]^2 * D[r[\[Theta]],\[Theta]]==-b*Cos[\[Theta]]; 
ic={}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to b \csc (\theta )+c_1 \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 8

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

\[ r{\left (t \right )} = C_{1} + \frac {b}{\sin {\left (t \right )}} \]