80.1.3 problem 4

Internal problem ID [18454]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter 1. section 5. Problems at page 19
Problem number : 4
Date solved : Thursday, March 13, 2025 at 12:00:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{x}+k^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+diff(y(x),x)/x+k^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (0, k x \right )+c_{2} \operatorname {BesselY}\left (0, k x \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 22

ode=D[y[x],{x,2}]+1/x*D[y[x],x]+k^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \operatorname {BesselJ}(0,k x)+c_2 \operatorname {BesselY}(0,k x) \]

✓ Sympy. Time used: 0.184 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(k**2*y(x) + Derivative(y(x), (x, 2)) + Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} J_{0}\left (k x\right ) + C_{2} Y_{0}\left (k x\right ) \]