80.5.12 problem B 11

Internal problem ID [21233]
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 11
Date solved : Thursday, October 02, 2025 at 07:27:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+b x^{\prime }+c x&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 47

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

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

✓ Sympy. Time used: 0.140 (sec). Leaf size: 39

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

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