86.9.3 problem 11

Internal problem ID [23173]
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 : 11
Date solved : Thursday, October 02, 2025 at 09:23:48 PM
CAS classification : [_separable]

\begin{align*} \left (1+\cos \left (\theta \right )\right ) r^{\prime }&=-r \sin \left (\theta \right ) \end{align*}

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

ode:=(1+cos(theta))*diff(r(theta),theta) = -r(theta)*sin(theta); 
dsolve(ode,r(theta), singsol=all);
 

\[ r = c_1 \left (1+\cos \left (\theta \right )\right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 21

ode=(1+Cos[\[Theta]])*D[r[\[Theta]],\[Theta]]==-r[\[Theta]]*Sin[\[Theta]]; 
ic={}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to c_1 \cos ^2\left (\frac {\theta }{2}\right )\\ r(\theta )&\to 0 \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
r = Function("r") 
ode = Eq((cos(t) + 1)*Derivative(r(t), t) + r(t)*sin(t),0) 
ics = {} 
dsolve(ode,func=r(t),ics=ics)
 

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