80.11.8 problem 8

Internal problem ID [21416]
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 12. Stability theory. Excercise 12.6 at page 270
Problem number : 8
Date solved : Thursday, October 02, 2025 at 07:30:58 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+2 h x^{\prime }+k^{2} x&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

ode:=diff(diff(x(t),t),t)+2*h*diff(x(t),t)+k^2*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 50

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

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

✓ Sympy. Time used: 0.131 (sec). Leaf size: 36

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

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