90.20.13 problem 15

Internal problem ID [25308]
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 337
Problem number : 15
Date solved : Thursday, October 02, 2025 at 11:59:51 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 43

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

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

✓ Mathematica. Time used: 0.129 (sec). Leaf size: 73

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

\begin{align*} y(t)&\to e^{-1/t} t \int _1^t-e^{\frac {1}{K[1]}-K[1]} (K[1]-1) K[1]dK[1]-e^{-t} \left (t^3+t^2-2\right )+c_1 e^{-1/t} t+c_2 (t-1) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(t), t) - (-t**2*exp(t)*Derivative(y(t), (t, 2)) + t