80.6.26 problem 26

Internal problem ID [21316]
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 : 26
Date solved : Thursday, October 02, 2025 at 07:28:23 PM
CAS classification : [[_3rd_order, _missing_y]]

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

With initial conditions

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 13

ode:=diff(diff(diff(x(t),t),t),t)-diff(x(t),t) = t; 
ic:=[x(0) = 0, D(x)(0) = 0, (D@@2)(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -\frac {t^{2}}{2}-1+\cosh \left (t \right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 31

ode=D[x[t],{t,3}]-D[x[t],t]==t; 
ic={x[0]==0,Derivative[1][x][0] ==0,Derivative[2][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} e^{-t} \left (-e^t \left (t^2+2\right )+e^{2 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 20

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

\[ x{\left (t \right )} = - \frac {t^{2}}{2} + \frac {e^{t}}{2} - 1 + \frac {e^{- t}}{2} \]