80.7.25 problem C 3

Internal problem ID [21344]
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 3
Date solved : Thursday, October 02, 2025 at 07:28:38 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+3 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.071 (sec). Leaf size: 44

ode:=[diff(x(t),t) = x(t), diff(y(t),t) = 2*y(t)+z(t), diff(z(t),t) = x(t)+3*z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 76

ode={D[x[t],t]==x[t],D[y[t],t]==2*y[t]+z[t],D[z[t],t]==x[t]+3*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 \frac {1}{2} e^t \left (c_1 \left (e^t-1\right )^2+2 e^t \left (c_3 \left (e^t-1\right )+c_2\right )\right )\\ z(t)&\to \left (\frac {c_1}{2}+c_3\right ) e^{3 t}-\frac {c_1 e^t}{2} \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 44

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) - 3*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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