80.5.8 problem B 6

Internal problem ID [21229]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 5. Second order equations. Excercise 5.9 at page 119
Problem number : B 6
Date solved : Thursday, October 02, 2025 at 07:27:06 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 4 x^{\prime }+2 x^{\prime \prime }&=-5 x \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 20

ode:=4*diff(x(t),t)+2*diff(diff(x(t),t),t) = -5*x(t); 
ic:=[x(0) = 0, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\sqrt {6}\, {\mathrm e}^{-t} \sin \left (\frac {\sqrt {6}\, t}{2}\right )}{3} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 28

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

\begin{align*} x(t)&\to \sqrt {\frac {2}{3}} e^{-t} \sin \left (\sqrt {\frac {3}{2}} t\right ) \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(5*x(t) + 4*Derivative(x(t), t) + 2*Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\sqrt {6} e^{- t} \sin {\left (\frac {\sqrt {6} t}{2} \right )}}{3} \]