59.1.470 problem 485

Internal problem ID [9642]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 485
Date solved : Wednesday, March 05, 2025 at 07:52:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 53

ode:=x^2*(x^2+2)*diff(diff(y(x),x),x)+2*x*(x^2+5)*diff(y(x),x)+2*(-x^2+3)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-c_{2} \sqrt {x^{2}+2}\, \left (x +2\right ) \left (x -2\right ) \sqrt {2}+x^{2} \left (\operatorname {arctanh}\left (\frac {\sqrt {2}}{\sqrt {x^{2}+2}}\right ) c_{2} +c_{1} \right ) \left (x^{2}+8\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.454 (sec). Leaf size: 112

ode=x^2*(2+x^2)*D[y[x],{x,2}]+2*x*(x^2+5)*D[y[x],x]+2*(3-x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (x^2+8\right ) \exp \left (\int _1^x\frac {3}{K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {2 K[2]^2+10}{K[2]^3+2 K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\frac {3}{K[1]^3+2 K[1]}dK[1]\right )}{\left (K[3]^2+8\right )^2}dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + 2*x*(x**2 + 5)*Derivative(y(x), x) + (6 - 2*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False