53.1.575 problem 591

Internal problem ID [11047]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 591
Date solved : Tuesday, September 30, 2025 at 07:35:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x \left (-x^{2}+4\right ) y^{\prime }+\left (7 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 24

ode:=4*x^2*diff(diff(y(x),x),x)+2*x*(-x^2+4)*diff(y(x),x)+(7*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{4}-16 x^{2}+32\right ) \left (c_1 +2 c_2 \right )}{32 \sqrt {x}} \]

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 70

ode=4*x^2*D[y[x],{x,2}]+2*x*(4-x^2)*D[y[x],x]+(1+7*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt {e} \left (x^4-16 x^2+32\right ) \left (c_2 \int _1^x\frac {e^{\frac {K[1]^2}{4}-1}}{K[1] \left (K[1]^4-16 K[1]^2+32\right )^2}dK[1]+c_1\right )}{\sqrt {x}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 2*x*(4 - x**2)*Derivative(y(x), x) + (7*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False