80.11.6 problem 6

Internal problem ID [21414]
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 12. Stability theory. Excercise 12.6 at page 270
Problem number : 6
Date solved : Thursday, October 02, 2025 at 07:30:57 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+2 x^{\prime }-x&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

ode:=diff(diff(x(t),t),t)+2*diff(x(t),t)-x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \left (c_1 \,{\mathrm e}^{2 t \sqrt {2}}+c_2 \right ) {\mathrm e}^{-\left (1+\sqrt {2}\right ) t} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 34

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

\begin{align*} x(t)&\to e^{-\left (\left (1+\sqrt {2}\right ) t\right )} \left (c_2 e^{2 \sqrt {2} t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 26

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

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