63.8.2 problem 2

Internal problem ID [13056]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.2.4. Applications. Exercises page 99
Problem number : 2
Date solved : Monday, March 31, 2025 at 07:33:02 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+\frac {x^{\prime }}{8}+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: 0.109 (sec). Leaf size: 31

ode:=diff(diff(x(t),t),t)+1/8*diff(x(t),t)+x(t) = 0; 
ic:=x(0) = 2, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {2 \,{\mathrm e}^{-\frac {t}{16}} \left (\sqrt {255}\, \sin \left (\frac {\sqrt {255}\, t}{16}\right )+255 \cos \left (\frac {\sqrt {255}\, t}{16}\right )\right )}{255} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 47

ode=D[x[t],{t,2}]+125/1000*D[x[t],t]+x[t]==0; 
ic={x[0]==2,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {2}{255} e^{-t/16} \left (\sqrt {255} \sin \left (\frac {\sqrt {255} t}{16}\right )+255 \cos \left (\frac {\sqrt {255} t}{16}\right )\right ) \]

✓ Sympy. Time used: 0.207 (sec). Leaf size: 37

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

\[ x{\left (t \right )} = \left (\frac {2 \sqrt {255} \sin {\left (\frac {\sqrt {255} t}{16} \right )}}{255} + 2 \cos {\left (\frac {\sqrt {255} t}{16} \right )}\right ) e^{- \frac {t}{16}} \]