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 : Tuesday, March 04, 2025 at 11:26:31 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*}

✓ Maple. Time used: 0.185 (sec). Leaf size: 18

ode:=diff(diff(diff(diff(x(t),t),t),t),t)+8*diff(diff(x(t),t),t)+16*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');
 

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

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

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

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

✓ Sympy. Time used: 0.144 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(16*x(t) + 8*Derivative(x(t), (t, 2)) + 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)
 

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