30.12.30 problem 39

Internal problem ID [7622]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 4, Linear Second-Order Equations. EXERCISES 4.2 at page 164
Problem number : 39
Date solved : Tuesday, September 30, 2025 at 04:55:04 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} z^{\prime \prime \prime }+2 z^{\prime \prime }-4 z^{\prime }-8 z&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=diff(diff(diff(z(t),t),t),t)+2*diff(diff(z(t),t),t)-4*diff(z(t),t)-8*z(t) = 0; 
dsolve(ode,z(t), singsol=all);
 

\[ z = \left (c_1 \,{\mathrm e}^{4 t}+c_3 t +c_2 \right ) {\mathrm e}^{-2 t} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 26

ode=D[z[t],{t,3}]+2*D[z[t],{t,2}]-4*D[z[t],{t,1}]-8*z[t]==0; 
ic={}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\begin{align*} z(t)&\to e^{-2 t} \left (c_2 t+c_3 e^{4 t}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(-8*z(t) - 4*Derivative(z(t), t) + 2*Derivative(z(t), (t, 2)) + Derivative(z(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=z(t),ics=ics)
 

\[ z{\left (t \right )} = C_{3} e^{2 t} + \left (C_{1} + C_{2} t\right ) e^{- 2 t} \]