59.3.2 problem Kovacic 1985 paper. page 14. section 3.2, example 2

Internal problem ID [10004]
Book : Collection of Kovacic problems
Section : section 3. Problems from Kovacic related papers
Problem number : Kovacic 1985 paper. page 14. section 3.2, example 2
Date solved : Sunday, March 30, 2025 at 02:51:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\left (\frac {6}{x^{2}}-1\right ) y \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 41

ode:=diff(diff(y(x),x),x) = (6/x^2-1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (c_1 \,x^{2}+3 c_2 x -3 c_1 \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_2 \,x^{2}-3 c_1 x -3 c_2 \right )}{x^{2}} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 21

ode=D[y[x],{x,2}]== ( (4*(5/2)^2-1)/(4*x^2)-1)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to x (c_1 j_2(x)-c_2 y_2(x)) \]

✓ Sympy. Time used: 0.094 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(-1 + 6/x**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {5}{2}}\left (x\right ) + C_{2} Y_{\frac {5}{2}}\left (x\right )\right ) \]