52.10.25 problem 26

Internal problem ID [8419]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 26
Date solved : Sunday, March 30, 2025 at 01:04:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-y \left (t \right )+z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 38

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 96

ode={D[x[t],t]==x[t],D[y[t],t]==3*y[t]+z[t],D[z[t],t]==-y[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^{2 t} (c_2 (t+1)+c_3 t) \\ z(t)\to e^{2 t} (c_3-(c_2+c_3) t) \\ x(t)\to 0 \\ y(t)\to e^{2 t} (c_2 (t+1)+c_3 t) \\ z(t)\to e^{2 t} (c_3-(c_2+c_3) t) \\ \end{align*}

✓ Sympy. Time used: 0.129 (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(-3*y(t) - z(t) + Derivative(y(t), t),0),Eq(y(t) - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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