85.53.8 problem 2 (b)

Internal problem ID [22821]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 197
Problem number : 2 (b)
Date solved : Thursday, October 02, 2025 at 09:14:58 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} s^{\prime \prime }+s^{\prime }&=t +{\mathrm e}^{-t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 24

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

\[ s = 2+\left (-t -2\right ) {\mathrm e}^{-t}+\frac {t^{2}}{2}-t \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 27

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

\begin{align*} s(t)&\to \frac {1}{2} \left (t^2-2 t-2 e^{-t} (t+2)+4\right ) \end{align*}

✓ Sympy. Time used: 0.137 (sec). Leaf size: 19

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

\[ s{\left (t \right )} = \frac {t^{2}}{2} - t + \left (- t - 2\right ) e^{- t} + 2 \]