59.1.783 problem 805

Internal problem ID [9955]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 805
Date solved : Wednesday, March 05, 2025 at 08:01:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 47

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

\[ y = \frac {c_{2} \left (x^{2}-1\right ) \ln \left (x -1\right )+\left (-x^{2}+1\right ) c_{2} \ln \left (x +1\right )+2 c_{1} x^{2}-2 c_{2} x -2 c_{1}}{2 x^{2}} \]

✓ Mathematica. Time used: 0.291 (sec). Leaf size: 81

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

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

✗ Sympy

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

False