80.6.27 problem 27

Internal problem ID [21317]
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 : 27
Date solved : Thursday, October 02, 2025 at 07:28:23 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} x^{\prime \prime \prime \prime }+x^{\prime \prime \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: 0.015 (sec). Leaf size: 31

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

\[ x = -{\mathrm e}^{-t} c_4 +\frac {\left (t^{2}-2 t +2\right ) c_4}{2}+\frac {t^{3} \left (t -4\right )}{24} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 42

ode=D[x[t],{t,4}]-D[x[t],{t,3}]==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 {t^4}{24}-\frac {t^3}{6}-\frac {c_1 t^2}{2}-c_1 t+c_1 \left (e^t-1\right ) \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t - Derivative(x(t), (t, 3)) + Derivative(x(t), (t, 4)),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)
 

\[ x{\left (t \right )} = - \frac {C_{4} t^{2}}{2} - C_{4} t + C_{4} e^{t} - C_{4} - \frac {t^{4}}{24} - \frac {t^{3}}{6} \]