82.7.12 problem 34-18

Internal problem ID [21868]
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 : 34-18
Date solved : Thursday, October 02, 2025 at 08:03:05 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= -\frac {2 c_2 \,{\mathrm e}^{4 t}}{3}+c_1 \,{\mathrm e}^{t} \\ y \left (t \right ) &= c_2 \,{\mathrm e}^{4 t}+c_3 \,{\mathrm e}^{-2 t} \\ z \left (t \right ) &= c_2 \,{\mathrm e}^{4 t}-c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 96

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

\begin{align*} x(t)&\to -\frac {1}{3} e^t \left ((c_2+c_3) \left (e^{3 t}-1\right )-3 c_1\right )\\ y(t)&\to \frac {1}{2} e^{-2 t} \left (c_2 \left (e^{6 t}+1\right )+c_3 \left (e^{6 t}-1\right )\right )\\ z(t)&\to \frac {1}{2} e^{-2 t} \left (c_2 \left (e^{6 t}-1\right )+c_3 \left (e^{6 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.072 (sec). Leaf size: 48

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

\[ \left [ x{\left (t \right )} = C_{1} e^{t} - \frac {2 C_{2} e^{4 t}}{3}, \ y{\left (t \right )} = C_{2} e^{4 t} - C_{3} e^{- 2 t}, \ z{\left (t \right )} = C_{2} e^{4 t} + C_{3} e^{- 2 t}\right ] \]