68.20.5 problem 5

Internal problem ID [17924]
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 : 5
Date solved : Thursday, October 02, 2025 at 02:29:51 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+13 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 20

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+13*x(t) = 0; 
ic:=[x(0) = 1, D(x)(0) = -1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{-2 t} \left (\frac {\sin \left (3 t \right )}{3}+\cos \left (3 t \right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=D[x[t],{t,2}]+4*D[x[t],t]+13*x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==-1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{-2 t} (\sin (3 t)+3 \cos (3 t)) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(13*x(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): -1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {\sin {\left (3 t \right )}}{3} + \cos {\left (3 t \right )}\right ) e^{- 2 t} \]