2.10.5 problem 19

Internal problem ID [866]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.4, Mechanical Vibrations. Page 337
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 04:18:47 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 x^{\prime \prime }+20 x^{\prime }+169 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ x^{\prime }\left (0\right )&=16 \\ \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 22

ode:=4*diff(diff(x(t),t),t)+20*diff(x(t),t)+169*x(t) = 0; 
ic:=[x(0) = 4, D(x)(0) = 16]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{-\frac {5 t}{2}} \left (\frac {13 \sin \left (6 t \right )}{3}+4 \cos \left (6 t \right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 29

ode=4*D[x[t],{t,2}]+20*D[x[t],t]+169*x[t]==0; 
ic={x[0]==4,Derivative[1][x][0 ]==16}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{-5 t/2} (13 \sin (6 t)+12 \cos (6 t)) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(169*x(t) + 20*Derivative(x(t), t) + 4*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 4, Subs(Derivative(x(t), t), t, 0): 16} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {13 \sin {\left (6 t \right )}}{3} + 4 \cos {\left (6 t \right )}\right ) e^{- \frac {5 t}{2}} \]