90.25.14 problem 14

Internal problem ID [25395]
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 379
Problem number : 14
Date solved : Friday, October 03, 2025 at 12:01:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

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

\[ y = c_2 t -c_2 +{\mathrm e}^{-t} c_1 -\frac {t^{2} {\mathrm e}^{-t}}{2} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 29

ode=t*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^{-t} \left (-\frac {t^2}{2}+1+c_1\right )+c_2 (t-1) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*exp(-t) + t*Derivative(y(t), (t, 2)) + (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 - t*exp(t)*Derivative(y(t), (t, 2))