65.6.8 problem 12.1 (viii)

Internal problem ID [13678]
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 (viii)
Date solved : Wednesday, March 05, 2025 at 10:11:44 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 z^{\prime \prime }+7 z^{\prime }-4 z&=0 \end{align*}

With initial conditions

\begin{align*} z \left (0\right )&=0\\ z^{\prime }\left (0\right )&=9 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 16

ode:=2*diff(diff(z(t),t),t)+7*diff(z(t),t)-4*z(t) = 0; 
ic:=z(0) = 0, D(z)(0) = 9; 
dsolve([ode,ic],z(t), singsol=all);
 

\[ z = 2 \left ({\mathrm e}^{\frac {9 t}{2}}-1\right ) {\mathrm e}^{-4 t} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 49

ode=D[z[t],{t,2}]+7*D[z[t],t]-4*z[t]==0; 
ic={z[0]==3,Derivative[1][z][0]==9}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\[ z(t)\to \frac {3}{10} e^{-\frac {1}{2} \left (7+\sqrt {65}\right ) t} \left (\left (5+\sqrt {65}\right ) e^{\sqrt {65} t}+5-\sqrt {65}\right ) \]

✓ Sympy. Time used: 0.179 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(-4*z(t) + 7*Derivative(z(t), t) + 2*Derivative(z(t), (t, 2)),0) 
ics = {z(0): 0, Subs(Derivative(z(t), t), t, 0): 9} 
dsolve(ode,func=z(t),ics=ics)
 

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