59.1.572 problem 588

Internal problem ID [9744]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 588
Date solved : Sunday, March 30, 2025 at 02:45:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)-x*(9*x^2+1)*diff(y(x),x)+(25*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 \left (x^{4}-4 x^{2}+1\right ) \ln \left (x \right )+c_1 \,x^{4}+\left (-4 c_1 -6 c_2 \right ) x^{2}+c_1 +3 c_2 \right ) x \]

✓ Mathematica. Time used: 0.609 (sec). Leaf size: 138

ode=x^2*(1+x^2)*D[y[x],{x,2}]-x*(1+9*x^2)*D[y[x],x]+(1+25*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (x^4-4 x^2+1\right ) \exp \left (\int _1^x\frac {1-7 K[1]^2}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x-\frac {9 K[2]^2+1}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\frac {1-7 K[1]^2}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )}{\left (K[3]^4-4 K[3]^2+1\right )^2}dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) - x*(9*x**2 + 1)*Derivative(y(x), x) + (25*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False