7.23.12 problem 18

Internal problem ID [598]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 18
Date solved : Saturday, March 29, 2025 at 04:57:37 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.132 (sec). Leaf size: 60

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

\begin{align*} x \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-4 t}}{2}+\frac {2 c_3 \,{\mathrm e}^{3 t}}{3}+\frac {c_1}{6} \\ y \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{-4 t}+c_3 \,{\mathrm e}^{3 t} \\ z \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{-4 t}}{2}-\frac {2 c_3 \,{\mathrm e}^{3 t}}{3}-\frac {13 c_1}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 190

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

\begin{align*} x(t)\to \frac {1}{84} e^{-4 t} \left (c_1 \left (-7 e^{4 t}+64 e^{7 t}+27\right )+24 c_2 \left (e^{7 t}-1\right )+c_3 \left (-7 e^{4 t}+16 e^{7 t}-9\right )\right ) \\ y(t)\to \frac {1}{14} e^{-4 t} \left (c_1 \left (-7 e^{4 t}+16 e^{7 t}-9\right )+c_2 \left (6 e^{7 t}+8\right )+c_3 \left (-7 e^{4 t}+4 e^{7 t}+3\right )\right ) \\ z(t)\to \frac {1}{84} e^{-4 t} \left (c_1 \left (91 e^{4 t}-64 e^{7 t}-27\right )-24 c_2 \left (e^{7 t}-1\right )-c_3 \left (-91 e^{4 t}+16 e^{7 t}-9\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 63

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1}}{13} - C_{2} e^{- 4 t} - C_{3} e^{3 t}, \ y{\left (t \right )} = - \frac {6 C_{1}}{13} + 2 C_{2} e^{- 4 t} - \frac {3 C_{3} e^{3 t}}{2}, \ z{\left (t \right )} = C_{1} + C_{2} e^{- 4 t} + C_{3} e^{3 t}\right ] \]