46.10.22 problem 23

Internal problem ID [9702]
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 : 23
Date solved : Tuesday, September 30, 2025 at 06:32:07 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.169 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 86

ode={D[x[t],t]==3*x[t]-y[t]-z[t],D[y[t],t]==x[t]+y[t]-z[t],D[z[t],t]==x[t]-y[t]+z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.066 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) + y(t) + z(t) + Derivative(x(t), t),0),Eq(-x(t) - y(t) + z(t) + Derivative(y(t), t),0),Eq(-x(t) + y(t) - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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