44.29.9 problem 3(e)

Internal problem ID [9491]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 3(e)
Date solved : Tuesday, September 30, 2025 at 06:19:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+2 y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )-y \left (t \right )+3 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: 0.117 (sec). Leaf size: 62

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{t}+c_3 \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -\frac {7 c_1 \,{\mathrm e}^{-t}}{3}-c_2 \,{\mathrm e}^{t}-\frac {c_3 \,{\mathrm e}^{3 t}}{5} \\ z \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{-t}}{3}+\frac {2 c_3 \,{\mathrm e}^{3 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 180

ode={D[x[t],t]==3*x[t]+2*y[t]+z[t],D[y[t],t]==-2*x[t]-y[t]+3*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 \frac {1}{8} e^{-t} \left (c_1 \left (6 e^{2 t}+5 e^{4 t}-3\right )+\left (e^{2 t}-1\right ) \left (c_2 \left (5 e^{2 t}+3\right )+2 c_3 \left (5 e^{2 t}-3\right )\right )\right )\\ y(t)&\to \frac {1}{8} e^{-t} \left (-\left (c_1 \left (6 e^{2 t}+e^{4 t}-7\right )\right )+c_2 \left (2 e^{2 t}-e^{4 t}+7\right )-2 c_3 \left (-8 e^{2 t}+e^{4 t}+7\right )\right )\\ z(t)&\to \frac {1}{4} e^{-t} \left (c_1 \left (e^{4 t}-1\right )+c_2 \left (e^{4 t}-1\right )+2 c_3 \left (e^{4 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) - 2*y(t) - z(t) + Derivative(x(t), t),0),Eq(2*x(t) + y(t) - 3*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)
 

\[ \left [ x{\left (t \right )} = \frac {3 C_{1} e^{- t}}{2} - C_{2} e^{t} + \frac {5 C_{3} e^{3 t}}{2}, \ y{\left (t \right )} = - \frac {7 C_{1} e^{- t}}{2} + C_{2} e^{t} - \frac {C_{3} e^{3 t}}{2}, \ z{\left (t \right )} = C_{1} e^{- t} + C_{3} e^{3 t}\right ] \]