86.9.10 problem 18

Internal problem ID [23180]
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 : 18
Date solved : Thursday, October 02, 2025 at 09:24:02 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

ode:=diff(r(theta),theta)*(sin(theta)-m*cos(theta))+r(theta)*(cos(theta)+m*sin(theta)) = 0; 
dsolve(ode,r(theta), singsol=all);
 

\[ r = \frac {c_1}{\sin \left (\theta \right )-m \cos \left (\theta \right )} \]

✓ Mathematica. Time used: 0.125 (sec). Leaf size: 24

ode=D[r[\[Theta]],\[Theta]]*(Sin[\[Theta]]-m*Cos[\[Theta]]) + r[\[Theta]]*(Cos[\[Theta]]+m*Sin[\[Theta]])==0; 
ic={}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to \frac {c_1}{m \cos (\theta )-\sin (\theta )}\\ r(\theta )&\to 0 \end{align*}

✓ Sympy. Time used: 0.303 (sec). Leaf size: 12

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

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