63.14.6 problem 1(f)

Internal problem ID [13103]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.5 Higher order equations. Exercises page 130
Problem number : 1(f)
Date solved : Wednesday, March 05, 2025 at 09:17:23 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime }-8 x&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

ode:=diff(diff(diff(x(t),t),t),t)-8*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 52

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

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

✓ Sympy. Time used: 0.114 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-8*x(t) + Derivative(x(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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