59.1.326 problem 333

Internal problem ID [9498]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 333
Date solved : Sunday, March 30, 2025 at 02:36:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (4 x^{2}-25\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 43

ode:=4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(4*x^2-25)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-3 c_2 \left (i x -\frac {1}{3} x^{2}+1\right ) {\mathrm e}^{-i x}+3 \,{\mathrm e}^{i x} \left (i x +\frac {1}{3} x^{2}-1\right ) c_1}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 59

ode=4*x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(4*x^2-25)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left (\left (-c_2 x^2+3 c_1 x+3 c_2\right ) \cos (x)+\left (c_1 \left (x^2-3\right )+3 c_2 x\right ) \sin (x)\right )}{x^{5/2}} \]

✓ Sympy. Time used: 0.211 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (4*x**2 - 25)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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