53.1.207 problem 210

Internal problem ID [10679]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 210
Date solved : Tuesday, September 30, 2025 at 07:30:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.385 (sec). Leaf size: 54

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

\begin{align*} y(t)&\to \frac {e^{-t-1} \left (\int _1^te^{K[1]+1} c_1 K[1]dK[1]+c_2\right )}{t}\\ y(t)&\to \frac {c_2 e^{-t-1}}{t} \end{align*}

✓ Sympy. Time used: 0.548 (sec). Leaf size: 388

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

\[ \text {Solution too large to show} \]