85.67.6 problem 6

Internal problem ID [22897]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 216
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:16:23 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{\prime \prime \prime \prime }-x&=8 \,{\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 42

ode=D[x[t],{t,4}]-x[t]==8*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.060 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-x(t) + Derivative(x(t), (t, 4)) - 8*exp(-t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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