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 : Wednesday, February 05, 2025 at 03:45:42 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*}

✓ Solution by Maple

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

dsolve([diff(x(t),t$4)+x(t)=0,x(0) = 0, D(x)(0) = 0, (D@@2)(x)(0) = 0, (D@@3)(x)(0) = 1],x(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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