68.20.4 problem 4

Internal problem ID [17923]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.2, page 241
Problem number : 4
Date solved : Thursday, October 02, 2025 at 02:29:50 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 x^{\prime \prime }+2 x^{\prime }+8 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 20

ode:=4*diff(diff(x(t),t),t)+2*diff(x(t),t)+8*x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {8 \sqrt {31}\, {\mathrm e}^{-\frac {t}{4}} \sin \left (\frac {\sqrt {31}\, t}{4}\right )}{31} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 30

ode=4*D[x[t],{t,2}]+2*D[x[t],t]+8*x[t]==0; 
ic={x[0]==0,Derivative[1][x][0 ]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {8 e^{-t/4} \sin \left (\frac {\sqrt {31} t}{4}\right )}{\sqrt {31}} \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(8*x(t) + 2*Derivative(x(t), t) + 4*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {8 \sqrt {31} e^{- \frac {t}{4}} \sin {\left (\frac {\sqrt {31} t}{4} \right )}}{31} \]