80.5.10 problem B 8

Internal problem ID [21231]
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 8
Date solved : Thursday, October 02, 2025 at 07:27:07 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+x^{\prime }-\beta x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

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

\[ x = \left (c_1 \,{\mathrm e}^{t \sqrt {1+4 \beta }}+c_2 \right ) {\mathrm e}^{-\frac {\left (1+\sqrt {1+4 \beta }\right ) t}{2}} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 43

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

\begin{align*} x(t)&\to e^{-\frac {1}{2} \left (\sqrt {4 \beta +1}+1\right ) t} \left (c_2 e^{\sqrt {4 \beta +1} t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 36

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

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