86.6.14 problem 14

Internal problem ID [23147]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5b at page 77
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:23:29 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} z^{\prime \prime }+8 z^{\prime }+16 z&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=diff(diff(z(t),t),t)+8*diff(z(t),t)+16*z(t) = 0; 
dsolve(ode,z(t), singsol=all);
 

\[ z = {\mathrm e}^{-4 t} \left (c_2 t +c_1 \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 18

ode=D[z[t],{t,2}]+8*D[z[t],t]+16*z[t]==0; 
ic={}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\begin{align*} z(t)&\to e^{-4 t} (c_2 t+c_1) \end{align*}

✓ Sympy. Time used: 0.089 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(16*z(t) + 8*Derivative(z(t), t) + Derivative(z(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=z(t),ics=ics)
 

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