14.29.5 problem 5

Internal problem ID [2803]
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 : 5
Date solved : Tuesday, March 04, 2025 at 02:42:51 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-7 x \left (t \right )+y \left (t \right )-6 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=10 x \left (t \right )-4 y \left (t \right )+12 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=2 x \left (t \right )-y \left (t \right )+z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 73

ode:=[diff(x(t),t) = -7*x(t)+y(t)-6*z(t), diff(y(t),t) = 10*x(t)-4*y(t)+12*z(t), diff(z(t),t) = 2*x(t)-y(t)+z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 166

ode={D[x[t],t]==-7*x[t]+1*y[t]-6*z[t],D[y[t],t]==10*x[t]-4*y[t]+12*z[t],D[z[t],t]==2*x[t]-1*y[t]+1*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-5 t} \left (c_1 \left (-4 e^{2 t}+2 e^{3 t}+3\right )+\left (e^t-1\right ) \left (c_2 e^{2 t}-3 c_3 e^t-3 c_3\right )\right ) \\ y(t)\to e^{-5 t} \left (-2 c_1 \left (-4 e^{2 t}+e^{3 t}+3\right )-c_2 e^{3 t}+2 (c_2+3 c_3) e^{2 t}-6 c_3\right ) \\ z(t)\to e^{-5 t} \left (-2 c_1 \left (-2 e^{2 t}+e^{3 t}+1\right )-c_2 e^{3 t}+(c_2+3 c_3) e^{2 t}-2 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(7*x(t) - y(t) + 6*z(t) + Derivative(x(t), t),0),Eq(-10*x(t) + 4*y(t) - 12*z(t) + Derivative(y(t), t),0),Eq(-2*x(t) + y(t) - z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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