7.19.7 problem 33

Internal problem ID [547]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 33
Date solved : Tuesday, March 04, 2025 at 11:26:29 AM
CAS classification : [[_high_order, _missing_x]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.261 (sec). Leaf size: 41

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

\[ x \left (t \right ) = -\frac {\sqrt {2}\, \left (\sinh \left (\frac {\sqrt {2}\, t}{2}\right ) \cos \left (\frac {\sqrt {2}\, t}{2}\right )-\cosh \left (\frac {\sqrt {2}\, t}{2}\right ) \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right )}{2} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 66

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

\[ x(t)\to \frac {e^{-\frac {t}{\sqrt {2}}} \left (\left (e^{\sqrt {2} t}+1\right ) \sin \left (\frac {t}{\sqrt {2}}\right )-\left (e^{\sqrt {2} t}-1\right ) \cos \left (\frac {t}{\sqrt {2}}\right )\right )}{2 \sqrt {2}} \]

✓ Sympy. Time used: 0.267 (sec). Leaf size: 90

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

\[ x{\left (t \right )} = \left (\frac {\sqrt {2} \sin {\left (\frac {\sqrt {2} t}{2} \right )}}{4} - \frac {\sqrt {2} \cos {\left (\frac {\sqrt {2} t}{2} \right )}}{4}\right ) e^{\frac {\sqrt {2} t}{2}} + \left (\frac {\sqrt {2} \sin {\left (\frac {\sqrt {2} t}{2} \right )}}{4} + \frac {\sqrt {2} \cos {\left (\frac {\sqrt {2} t}{2} \right )}}{4}\right ) e^{- \frac {\sqrt {2} t}{2}} \]