2.10.4 problem 18

Internal problem ID [865]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.4, Mechanical Vibrations. Page 337
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 04:18:46 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 x^{\prime \prime }+12 x^{\prime }+50 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=-8 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 14

ode:=2*diff(diff(x(t),t),t)+12*diff(x(t),t)+50*x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = -8]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = -2 \,{\mathrm e}^{-3 t} \sin \left (4 t \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 16

ode=2*D[x[t],{t,2}]+12*D[x[t],t]+50*x[t]==0; 
ic={x[0]==0,Derivative[1][x][0 ]==-8}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -2 e^{-3 t} \sin (4 t) \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(50*x(t) + 12*Derivative(x(t), t) + 2*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): -8} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = - 2 e^{- 3 t} \sin {\left (4 t \right )} \]