74.20.2 problem 2

Internal problem ID [16563]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.2, page 241
Problem number : 2
Date solved : Monday, March 31, 2025 at 02:58:51 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \frac {x^{\prime \prime }}{32}+2 x^{\prime }+x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1\\ x^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 37

ode:=1/32*diff(diff(x(t),t),t)+2*diff(x(t),t)+x(t) = 0; 
ic:=x(0) = 1, D(x)(0) = 0; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x = \frac {{\mathrm e}^{-4 \left (8+\sqrt {62}\right ) t} \left (\left (31+4 \sqrt {62}\right ) {\mathrm e}^{8 t \sqrt {62}}+31-4 \sqrt {62}\right )}{62} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 50

ode=1/32*D[x[t],{t,2}]+2*D[x[t],t]+x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {1}{62} e^{-4 \left (8+\sqrt {62}\right ) t} \left (\left (31+4 \sqrt {62}\right ) e^{8 \sqrt {62} t}+31-4 \sqrt {62}\right ) \]

✓ Sympy. Time used: 0.241 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + 2*Derivative(x(t), t) + Derivative(x(t), (t, 2))/32,0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {1}{2} + \frac {2 \sqrt {62}}{31}\right ) e^{4 t \left (-8 + \sqrt {62}\right )} + \left (\frac {1}{2} - \frac {2 \sqrt {62}}{31}\right ) e^{- 4 t \left (\sqrt {62} + 8\right )} \]