59.1.699 problem 716

Internal problem ID [9871]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 716
Date solved : Sunday, March 30, 2025 at 02:48:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

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

\[ u = \frac {c_1 \sin \left (a x \right )+c_2 \cos \left (a x \right )}{x} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 42

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

\[ u(x)\to \frac {e^{-i a x} \left (2 c_1-\frac {i c_2 e^{2 i a x}}{a}\right )}{2 x} \]

✓ Sympy. Time used: 0.226 (sec). Leaf size: 24

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

\[ u{\left (x \right )} = \frac {C_{1} J_{\frac {1}{2}}\left (a x\right ) + C_{2} Y_{\frac {1}{2}}\left (a x\right )}{\sqrt {x}} \]