59.1.276 problem 279

Internal problem ID [9448]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 279
Date solved : Sunday, March 30, 2025 at 02:35:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 33

ode:=diff(diff(u(x),x),x)+4/x*diff(u(x),x)+a^2*u(x) = 0; 
dsolve(ode,u(x), singsol=all);
 

\[ u = \frac {\left (c_1 a x +c_2 \right ) \cos \left (x a \right )+\sin \left (x a \right ) \left (c_2 a x -c_1 \right )}{x^{3}} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 57

ode=D[u[x],{x,2}]+4/x*D[u[x],x]+a^2*u[x]==0; 
ic={}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\[ u(x)\to -\frac {\sqrt {\frac {2}{\pi }} ((a c_1 x+c_2) \cos (a x)+(a c_2 x-c_1) \sin (a x))}{x^{3/2} (a x)^{3/2}} \]

✓ Sympy. Time used: 0.232 (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)) + 4*Derivative(u(x), x)/x,0) 
ics = {} 
dsolve(ode,func=u(x),ics=ics)
 

\[ u{\left (x \right )} = \frac {C_{1} J_{\frac {3}{2}}\left (a x\right ) + C_{2} Y_{\frac {3}{2}}\left (a x\right )}{x^{\frac {3}{2}}} \]