80.6.9 problem 9

Internal problem ID [21299]
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 : 9
Date solved : Thursday, October 02, 2025 at 07:27:55 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime }+x^{\prime \prime }-2 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=0 \\ x^{\prime \prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 24

ode:=diff(diff(diff(x(t),t),t),t)+diff(diff(x(t),t),t)-2*x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 0, (D@@2)(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\left (-2 \sin \left (t \right )-\cos \left (t \right )\right ) {\mathrm e}^{-t}}{5}+\frac {{\mathrm e}^{t}}{5} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 28

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

\begin{align*} x(t)&\to \frac {1}{5} e^{-t} \left (e^{2 t}-2 \sin (t)-\cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.074 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-2*x(t) + Derivative(x(t), (t, 2)) + 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): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \frac {2 \sin {\left (t \right )}}{5} - \frac {\cos {\left (t \right )}}{5}\right ) e^{- t} + \frac {e^{t}}{5} \]