80.10.5 problem 28

Internal problem ID [21400]
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 11. Laplace transform. Excercise 11.7 at page 248
Problem number : 28
Date solved : Thursday, October 02, 2025 at 07:30:49 PM
CAS classification : [[_high_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.047 (sec). Leaf size: 5

ode:=diff(diff(diff(diff(x(t),t),t),t),t)+diff(diff(x(t),t),t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 1, (D@@2)(x)(0) = 0, (D@@3)(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t),method='laplace');
 

\[ x = t \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 6

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

\begin{align*} x(t)&\to t \end{align*}

✓ Sympy. Time used: 0.044 (sec). Leaf size: 3

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

\[ x{\left (t \right )} = t \]