30.12.18 problem 18

Internal problem ID [7610]
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 : 18
Date solved : Tuesday, September 30, 2025 at 04:54:54 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} z \left (0\right )&=0 \\ z^{\prime }\left (0\right )&=-3 \\ \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 17

ode:=diff(diff(z(t),t),t)-2*diff(z(t),t)-2*z(t) = 0; 
ic:=[z(0) = 0, D(z)(0) = -3]; 
dsolve([ode,op(ic)],z(t), singsol=all);
 

\[ z = -{\mathrm e}^{t} \sqrt {3}\, \sinh \left (t \sqrt {3}\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 38

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

\begin{align*} z(t)&\to -\frac {1}{2} \sqrt {3} e^{t-\sqrt {3} t} \left (e^{2 \sqrt {3} t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(-2*z(t) - 2*Derivative(z(t), t) + Derivative(z(t), (t, 2)),0) 
ics = {z(0): 1, Subs(Derivative(z(t), t), t, 0): -3} 
dsolve(ode,func=z(t),ics=ics)
 

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