90.24.16 problem 16

Internal problem ID [25381]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 371
Problem number : 16
Date solved : Friday, October 03, 2025 at 12:00:45 AM
CAS classification : [_Gegenbauer]

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

Using reduction of order method given that one solution is

\begin{align*} y&=t \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 33

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

\begin{align*} y(t)&\to c_1 t-\frac {1}{2} c_2 (t \log (1-t)-t \log (t+1)+2) \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)) + 2*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

False