80.6.18 problem 18

Internal problem ID [21308]
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 6. Higher order linear equations. Excercise 6.5 at page 133
Problem number : 18
Date solved : Thursday, October 02, 2025 at 07:28:17 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} x^{\left (5\right )}-x^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

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

\[ x = c_1 +c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t}+c_4 \sin \left (t \right )+c_5 \cos \left (t \right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 77

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

\begin{align*} x(t)&\to -\frac {c_2 e^{-\sqrt [4]{2} t}}{\sqrt [4]{2}}+\frac {c_4 e^{\sqrt [4]{2} t}}{\sqrt [4]{2}}-\frac {c_3 \cos \left (\sqrt [4]{2} t\right )}{\sqrt [4]{2}}+\frac {c_1 \sin \left (\sqrt [4]{2} t\right )}{\sqrt [4]{2}}+c_5 \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 44

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

\[ x{\left (t \right )} = C_{1} + C_{2} e^{- \sqrt [4]{2} t} + C_{3} e^{\sqrt [4]{2} t} + C_{4} \sin {\left (\sqrt [4]{2} t \right )} + C_{5} \cos {\left (\sqrt [4]{2} t \right )} \]