80.11.18 problem 16

Internal problem ID [21426]
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 : 16
Date solved : Thursday, October 02, 2025 at 07:31:27 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime }+5 x^{\prime \prime }+9 x^{\prime }+5 x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 22

ode:=diff(diff(diff(x(t),t),t),t)+5*diff(diff(x(t),t),t)+9*diff(x(t),t)+5*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 28

ode=D[x[t],{t,3}]+5*D[x[t],{t,2}]+9*D[x[t],t]+5*x[t]==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(5*x(t) + 9*Derivative(x(t), t) + 5*Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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