2.10.1 problem 15

Internal problem ID [862]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.4, Mechanical Vibrations. Page 337
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 04:18:44 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \frac {x^{\prime \prime }}{2}+3 x^{\prime }+4 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 17

ode:=1/2*diff(diff(x(t),t),t)+3*diff(x(t),t)+4*x(t) = 0; 
ic:=[x(0) = 2, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = 4 \,{\mathrm e}^{-2 t}-2 \,{\mathrm e}^{-4 t} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 20

ode=1/2*D[x[t],{t,2}]+3*D[x[t],t]+4*x[t]==0; 
ic={x[0]==2,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.118 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*x(t) + 3*Derivative(x(t), t) + Derivative(x(t), (t, 2))/2,0) 
ics = {x(0): 2, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

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