59.1.625 problem 642

Internal problem ID [9797]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 642
Date solved : Sunday, March 30, 2025 at 02:46:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

\[ y = \frac {c_2 \,{\mathrm e}^{t}+c_1 t +c_1}{t} \]

✓ Mathematica. Time used: 0.249 (sec). Leaf size: 80

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

\begin{align*} y(t)\to \exp \left (\int _1^t\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \left (\int _1^t\exp \left (-\int _1^{K[2]}\left (1-\frac {1}{K[1]}\right )dK[1]\right ) c_1dK[2]+c_2\right ) \\ y(t)\to c_2 \exp \left (\int _1^t\left (1-\frac {1}{K[1]}\right )dK[1]\right ) \\ \end{align*}

✓ Sympy. Time used: 1.099 (sec). Leaf size: 386

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

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