82.7.11 problem 34-17

Internal problem ID [21867]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-17
Date solved : Thursday, October 02, 2025 at 08:03:04 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 )+3 y \left (t \right )-z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=3 x \left (t \right )+3 y \left (t \right )-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 57

ode:=[diff(x(t),t) = 3*x(t)+y(t)-z(t), diff(y(t),t) = x(t)+3*y(t)-z(t), diff(z(t),t) = 3*x(t)+3*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 ) &= 2 c_2 \,{\mathrm e}^{2 t}+3 c_3 \,{\mathrm e}^{t}+{\mathrm e}^{2 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 104

ode={D[x[t],t]==3*x[t]+y[t]-z[t],D[y[t],t]==x[t]+3*y[t]-z[t],D[z[t],t]==3*x[t]+3*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_2 \left (2 e^t-1\right )-c_3 \left (e^t-1\right )\right )\\ z(t)&\to e^t \left (3 c_1 \left (e^t-1\right )+3 c_2 \left (e^t-1\right )+c_3 \left (3-2 e^t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 46

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

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