86.9.5 problem 13

Internal problem ID [23175]
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 : 13
Date solved : Saturday, October 04, 2025 at 05:35:55 PM
CAS classification : [_quadrature]

\begin{align*} r r^{\prime }&=a \end{align*}

With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.08 (sec). Leaf size: 35

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

\begin{align*} r(\theta )&\to -\sqrt {2 a \theta +b^2}\\ r(\theta )&\to \sqrt {2 a \theta +b^2} \end{align*}

✓ Sympy. Time used: 0.263 (sec). Leaf size: 14

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

\[ r{\left (t \right )} = \sqrt {2 a t + b^{2}} \]