59.1.397 problem 409

Internal problem ID [9569]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 409
Date solved : Wednesday, March 05, 2025 at 07:51:03 AM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 33

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

\[ y(x)\to c_1 x-\frac {1}{2} c_2 (x \log (1-x)-x \log (x+1)+2) \]

✗ Sympy

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

False