80.8.4 problem 4

Internal problem ID [18512]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter VII. Linear equations of order higher than the first. section 56. Problems at page 163
Problem number : 4
Date solved : Monday, March 31, 2025 at 05:40:54 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} r^{\prime \prime }-a^{2} r&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 18

ode:=diff(diff(r(phi),phi),phi)-a^2*r(phi) = 0; 
dsolve(ode,r(phi), singsol=all);
 

\[ r = c_1 \,{\mathrm e}^{-a \phi }+c_2 \,{\mathrm e}^{a \phi } \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 23

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

\[ r(\phi )\to c_1 e^{a \phi }+c_2 e^{-a \phi } \]

✓ Sympy. Time used: 0.084 (sec). Leaf size: 15

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

\[ r{\left (\phi \right )} = C_{1} e^{- a \phi } + C_{2} e^{a \phi } \]