2.10.6 problem 20

Internal problem ID [867]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.4, Mechanical Vibrations. Page 337
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 04:18:48 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 x^{\prime \prime }+16 x^{\prime }+40 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=5 \\ x^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 22

ode:=2*diff(diff(x(t),t),t)+16*diff(x(t),t)+40*x(t) = 0; 
ic:=[x(0) = 5, D(x)(0) = 4]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{-4 t} \left (12 \sin \left (2 t \right )+5 \cos \left (2 t \right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 24

ode=2*D[x[t],{t,2}]+16*D[x[t],t]+40*x[t]==0; 
ic={x[0]==5,Derivative[1][x][0 ]==4}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-4 t} (12 \sin (2 t)+5 \cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(40*x(t) + 16*Derivative(x(t), t) + 2*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 5, Subs(Derivative(x(t), t), t, 0): 4} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (12 \sin {\left (2 t \right )} + 5 \cos {\left (2 t \right )}\right ) e^{- 4 t} \]