80.7.30 problem C 9

Internal problem ID [21349]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 7. System of first order equations. Excercise 7.6 at page 162
Problem number : C 9
Date solved : Thursday, October 02, 2025 at 07:28:40 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=2 x\\ y^{\prime }\left (t \right )&=2 y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=-x-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.067 (sec). Leaf size: 52

ode:=[diff(x(t),t) = 2*x(t), diff(y(t),t) = 2*y(t)+z(t), diff(z(t),t) = -x(t)-z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_3 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t} t c_3}{3}+c_1 \,{\mathrm e}^{2 t}-\frac {c_2 \,{\mathrm e}^{-t}}{3} \\ z \left (t \right ) &= -\frac {c_3 \,{\mathrm e}^{2 t}}{3}+c_2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 88

ode={D[x[t],t]==2*x[t],D[y[t],t]==2*y[t]+z[t],D[z[t],t]==-x[t]-z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 e^{2 t}\\ y(t)&\to \frac {1}{9} e^{-t} \left (c_1 \left (e^{3 t} (1-3 t)-1\right )+3 (3 c_2+c_3) e^{3 t}-3 c_3\right )\\ z(t)&\to \frac {1}{3} e^{-t} \left (c_1 \left (-e^{3 t}\right )+c_1+3 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.067 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) + Derivative(x(t), t),0),Eq(-2*y(t) - z(t) + Derivative(y(t), t),0),Eq(x(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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