14.29.6 problem 6

Internal problem ID [2804]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 02:42:52 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 )+4 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+2 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=4 x \left (t \right )+2 y \left (t \right )+3 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 66

ode:=[diff(x(t),t) = 3*x(t)+2*y(t)+4*z(t), diff(y(t),t) = 2*x(t)+2*z(t), diff(z(t),t) = 4*x(t)+2*y(t)+3*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 2 c_2 \,{\mathrm e}^{8 t}+2 c_3 \,{\mathrm e}^{-t}+{\mathrm e}^{-t} c_1 \\ y &= c_2 \,{\mathrm e}^{8 t}+c_3 \,{\mathrm e}^{-t} \\ z \left (t \right ) &= 2 c_2 \,{\mathrm e}^{8 t}-\frac {5 c_3 \,{\mathrm e}^{-t}}{2}-{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 135

ode={D[x[t],t]==3*x[t]+2*y[t]+4*z[t],D[y[t],t]==2*x[t]-0*y[t]+2*z[t],D[z[t],t]==4*x[t]+2*y[t]+3*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{9} e^{-t} \left (c_1 \left (4 e^{9 t}+5\right )+2 (c_2+2 c_3) \left (e^{9 t}-1\right )\right ) \\ y(t)\to \frac {1}{9} e^{-t} \left (2 c_1 \left (e^{9 t}-1\right )+c_2 \left (e^{9 t}+8\right )+2 c_3 \left (e^{9 t}-1\right )\right ) \\ z(t)\to \frac {1}{9} e^{-t} \left (4 c_1 \left (e^{9 t}-1\right )+2 c_2 \left (e^{9 t}-1\right )+c_3 \left (4 e^{9 t}+5\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) - 2*y(t) - 4*z(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 2*z(t) + Derivative(y(t), t),0),Eq(-4*x(t) - 2*y(t) - 3*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{3} e^{8 t} - \left (C_{1} + \frac {C_{2}}{2}\right ) e^{- t}, \ y{\left (t \right )} = C_{2} e^{- t} + \frac {C_{3} e^{8 t}}{2}, \ z{\left (t \right )} = C_{1} e^{- t} + C_{3} e^{8 t}\right ] \]