86.9.1 problem 9

Internal problem ID [23171]
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 : 9
Date solved : Thursday, October 02, 2025 at 09:23:46 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} r \left (0\right )&=2 a \\ \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 10

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 11

ode=D[r[\[Theta]],\[Theta]]==-a*Sin[\[Theta]]; 
ic={r[0]==2*a}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to a (\cos (\theta )+1) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 8

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

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