80.5.11 problem B 10

Internal problem ID [21232]
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 10
Date solved : Thursday, October 02, 2025 at 07:27:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+k x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+k*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \left (c_1 \,{\mathrm e}^{2 t \sqrt {-k +4}}+c_2 \right ) {\mathrm e}^{-\left (2+\sqrt {-k +4}\right ) t} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 42

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

\begin{align*} x(t)&\to e^{-\left (\left (\sqrt {4-k}+2\right ) t\right )} \left (c_2 e^{2 \sqrt {4-k} t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 29

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

\[ x{\left (t \right )} = C_{1} e^{t \left (\sqrt {4 - k} - 2\right )} + C_{2} e^{- t \left (\sqrt {4 - k} + 2\right )} \]