86.9.6 problem 14

Internal problem ID [23176]
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 : 14
Date solved : Thursday, October 02, 2025 at 09:23:55 PM
CAS classification : [_separable]

\begin{align*} r^{\prime } \left (1+\frac {\cos \left (\theta \right )}{2}\right )-r \sin \left (\theta \right )&=0 \end{align*}

With initial conditions

\begin{align*} r \left (\frac {\pi }{2}\right )&=2 a \\ \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 13

ode:=diff(r(theta),theta)*(1+1/2*cos(theta))-r(theta)*sin(theta) = 0; 
ic:=[r(1/2*Pi) = 2*a]; 
dsolve([ode,op(ic)],r(theta), singsol=all);
 

\[ r = \frac {8 a}{\left (2+\cos \left (\theta \right )\right )^{2}} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 36

ode=D[r[\[Theta]],\[Theta]]*(1+1/2*Cos[\[Theta]] )-r[\[Theta]] ==0; 
ic={r[Pi/2]==2*a}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to 2 a e^{-\frac {2 \left (\pi -6 \arctan \left (\frac {\tan \left (\frac {\theta }{2}\right )}{\sqrt {3}}\right )\right )}{3 \sqrt {3}}} \end{align*}

✓ Sympy. Time used: 0.223 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
a = symbols("a") 
r = Function("r") 
ode = Eq((cos(t)/2 + 1)*Derivative(r(t), t) - r(t)*sin(t),0) 
ics = {r(pi/2): 2*a} 
dsolve(ode,func=r(t),ics=ics)
 

\[ r{\left (t \right )} = \frac {8 a}{\cos ^{2}{\left (t \right )} + 4 \cos {\left (t \right )} + 4} \]