59.1.564 problem 580

Internal problem ID [9736]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 580
Date solved : Sunday, March 30, 2025 at 02:45:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 35

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

\[ y = \frac {\left (-c_2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )+c_2 \sqrt {x^{2}+1}+c_1 \right ) x}{\left (x^{2}+1\right )^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 96

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

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

✗ Sympy

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

False