80.11.19 problem 17

Internal problem ID [21427]
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 : 17
Date solved : Thursday, October 02, 2025 at 07:31:27 PM
CAS classification : [[_high_order, _missing_x]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

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

\[ x = c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )+c_4 \,{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 64

ode=D[x[t],{t,4}]+D[x[t],{t,3}]-D[x[t],t]-x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} \left (c_4 e^{2 t}+c_2 e^{t/2} \cos \left (\frac {\sqrt {3} t}{2}\right )+c_1 e^{t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )+c_3\right ) \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-x(t) - Derivative(x(t), t) + Derivative(x(t), (t, 3)) + Derivative(x(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{3} e^{- t} + C_{4} e^{t} + \left (C_{1} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]