7.19.9 problem 35

Internal problem ID [549]
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 : 35
Date solved : Wednesday, February 05, 2025 at 03:45:44 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime \prime }+8 x^{\prime \prime }+16 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: 0.185 (sec). Leaf size: 18

dsolve([diff(x(t),t$4)+8*diff(x(t),t$2)+16*x(t)=0,x(0) = 0, D(x)(0) = 0, (D@@2)(x)(0) = 0, (D@@3)(x)(0) = 1],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\sin \left (2 t \right )}{16}-\frac {t \cos \left (2 t \right )}{8} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 21

DSolve[{D[x[t],{t,4}]+8*D[x[t],{t,2}]+16*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]
 

\[ x(t)\to \frac {1}{16} (\sin (2 t)-2 t \cos (2 t)) \]