53.1.616 problem 632

Internal problem ID [11088]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 632
Date solved : Tuesday, September 30, 2025 at 07:35:48 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-t^{2}+1\right ) y^{\prime \prime }-2 t y^{\prime }+6 y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 44

ode:=(-t^2+1)*diff(diff(y(t),t),t)-2*t*diff(y(t),t)+6*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 55

ode=(1-t^2)*D[y[t],{t,2}]-2*t*D[y[t],t]+6*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{2} c_1 \left (3 t^2-1\right )-\frac {1}{4} c_2 \left (\left (3 t^2-1\right ) \log (1-t)+\left (1-3 t^2\right ) \log (t+1)+6 t\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*Derivative(y(t), t) + (1 - t**2)*Derivative(y(t), (t, 2)) + 6*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

False