65.6.3 problem 12.1 (iii)

Internal problem ID [13673]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 12, Homogeneous second order linear equations. Exercises page 118
Problem number : 12.1 (iii)
Date solved : Wednesday, March 05, 2025 at 10:11:29 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} z^{\prime \prime }-4 z^{\prime }+13 z&=0 \end{align*}

With initial conditions

\begin{align*} z \left (0\right )&=7\\ z^{\prime }\left (0\right )&=42 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 23

ode:=diff(diff(z(t),t),t)-4*diff(z(t),t)+13*z(t) = 0; 
ic:=z(0) = 7, D(z)(0) = 42; 
dsolve([ode,ic],z(t), singsol=all);
 

\[ z = \frac {7 \,{\mathrm e}^{2 t} \left (4 \sin \left (3 t \right )+3 \cos \left (3 t \right )\right )}{3} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 27

ode=D[z[t],{t,2}]-4*D[z[t],t]+13*z[t]==0; 
ic={z[0]==7,Derivative[1][z][0]==42}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\[ z(t)\to \frac {7}{3} e^{2 t} (4 \sin (3 t)+3 \cos (3 t)) \]

✓ Sympy. Time used: 0.186 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(13*z(t) - 4*Derivative(z(t), t) + Derivative(z(t), (t, 2)),0) 
ics = {z(0): 7, Subs(Derivative(z(t), t), t, 0): 42} 
dsolve(ode,func=z(t),ics=ics)
 

\[ z{\left (t \right )} = \left (\frac {28 \sin {\left (3 t \right )}}{3} + 7 \cos {\left (3 t \right )}\right ) e^{2 t} \]