90.20.24 problem 17 (3 c)

Internal problem ID [25319]
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 : 17 (3 c)
Date solved : Friday, October 03, 2025 at 12:00:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 12

ode:=(t-1)*diff(diff(y(t),t),t)-t*diff(y(t),t)+y(t) = 2*t*exp(-t); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 3 t -2 \sinh \left (t \right ) \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 19

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

\begin{align*} y(t)&\to 3 t+e^{-t}-e^t \end{align*}

✗ Sympy

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

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