59.1.142 problem 144

Internal problem ID [9314]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 144
Date solved : Sunday, March 30, 2025 at 02:32:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.21 (sec). Leaf size: 112

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

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

✗ Sympy

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

False