80.5.56 problem D 6

Internal problem ID [21277]
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 : D 6
Date solved : Thursday, October 02, 2025 at 07:27:36 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-\frac {t}{2}} \left (c_1 \sin \left (\frac {\sqrt {3}\, t}{2}\right )+c_2 \cos \left (\frac {\sqrt {3}\, t}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 42

ode=D[x[t],{t,2}]+D[x[t],t]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t/2} \left (c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 31

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

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