80.6.6 problem 6

Internal problem ID [21296]
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 : 6
Date solved : Thursday, October 02, 2025 at 07:27:55 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime }+4 x^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ x^{\prime }\left (0\right )&=-1 \\ x^{\prime \prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 18

ode:=diff(diff(diff(x(t),t),t),t)+4*diff(x(t),t) = 0; 
ic:=[x(0) = 1, D(x)(0) = -1, (D@@2)(x)(0) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {3}{2}-\frac {\sin \left (2 t \right )}{2}-\frac {\cos \left (2 t \right )}{2} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 23

ode=D[x[t],{t,3}]+4*D[x[t],t]==0; 
ic={x[0]==1,Derivative[1][x][0] ==-1,Derivative[2][x][0] ==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} (-\sin (2 t)-\cos (2 t)+3) \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*Derivative(x(t), t) + Derivative(x(t), (t, 3)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): -1, Subs(Derivative(x(t), (t, 2)), t, 0): 2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = - \frac {\sin {\left (2 t \right )}}{2} - \frac {\cos {\left (2 t \right )}}{2} + \frac {3}{2} \]