80.7.27 problem C 5

Internal problem ID [21346]
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 5
Date solved : Thursday, October 02, 2025 at 07:28:39 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=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: 0.069 (sec). Leaf size: 45

ode:=[diff(x(t),t) = 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}^{t} \\ y \left (t \right ) &= -c_3 t \,{\mathrm e}^{t}-c_2 \,{\mathrm e}^{t}-c_3 \,{\mathrm e}^{t}+c_1 \,{\mathrm e}^{2 t} \\ z \left (t \right ) &= \left (c_3 t +c_2 \right ) {\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 57

ode={D[x[t],t]==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^t\\ y(t)&\to e^t \left (c_1 \left (-t+e^t-1\right )+(c_2+c_3) e^t-c_3\right )\\ z(t)&\to e^t (c_1 t+c_3) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-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)
 

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