86.5.18 problem 18

Internal problem ID [23132]
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 5a at page 74
Problem number : 18
Date solved : Thursday, October 02, 2025 at 09:23:14 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} z^{\prime \prime }-3 z^{\prime }+z&=0 \end{align*}

With initial conditions

\begin{align*} z \left (0\right )&=1 \\ z^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 36

ode:=diff(diff(z(t),t),t)-3*diff(z(t),t)+z(t) = 0; 
ic:=[z(0) = 1, D(z)(0) = 0]; 
dsolve([ode,op(ic)],z(t), singsol=all);
 

\[ z = \frac {\left (\left (5-3 \sqrt {5}\right ) {\mathrm e}^{t \sqrt {5}}+3 \sqrt {5}+5\right ) {\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}}}{10} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 51

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

\begin{align*} z(t)&\to \frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (\left (5-3 \sqrt {5}\right ) e^{\sqrt {5} t}+5+3 \sqrt {5}\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 49

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

\[ z{\left (t \right )} = \left (\frac {1}{2} + \frac {3 \sqrt {5}}{10}\right ) e^{\frac {t \left (3 - \sqrt {5}\right )}{2}} + \left (\frac {1}{2} - \frac {3 \sqrt {5}}{10}\right ) e^{\frac {t \left (\sqrt {5} + 3\right )}{2}} \]