80.5.29 problem C 5

Internal problem ID [21250]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 5. Second order equations. Excercise 5.9 at page 119
Problem number : C 5
Date solved : Thursday, October 02, 2025 at 07:27:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime }-x&={\mathrm e}^{-3 t} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

ode:=diff(diff(x(t),t),t)-x(t) = exp(-3*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-t} c_2 +{\mathrm e}^{t} c_1 +\frac {{\mathrm e}^{-3 t}}{8} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 29

ode=D[x[t],{t,2}]-x[t]==Exp[-3*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {e^{-3 t}}{8}+c_1 e^t+c_2 e^{-t} \end{align*}

✓ Sympy. Time used: 0.046 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-x(t) + Derivative(x(t), (t, 2)) - exp(-3*t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{t} + \frac {e^{- 3 t}}{8} \]