52.10.13 problem 12

Internal problem ID [8407]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 12
Date solved : Sunday, March 30, 2025 at 01:00:59 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.194 (sec). Leaf size: 57

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-5 t} c_2 +{\mathrm e}^{3 t} c_1 +\frac {2 c_3 \,{\mathrm e}^{6 t}}{11} \\ y \left (t \right ) &= -\frac {2 c_3 \,{\mathrm e}^{6 t}}{11}-{\mathrm e}^{-5 t} c_2 +{\mathrm e}^{3 t} c_1 \\ z \left (t \right ) &= c_3 \,{\mathrm e}^{6 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 105

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

\begin{align*} x(t)\to \frac {1}{22} e^{-5 t} \left (11 c_1 \left (e^{8 t}+1\right )+11 c_2 \left (e^{8 t}-1\right )+4 c_3 \left (e^{11 t}-1\right )\right ) \\ y(t)\to \frac {1}{22} e^{-5 t} \left (11 c_1 \left (e^{8 t}-1\right )+11 c_2 \left (e^{8 t}+1\right )-4 c_3 \left (e^{11 t}-1\right )\right ) \\ z(t)\to c_3 e^{6 t} \\ \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 60

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

\[ \left [ x{\left (t \right )} = - C_{1} e^{- 5 t} + C_{2} e^{3 t} + \frac {2 C_{3} e^{6 t}}{11}, \ y{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{3 t} - \frac {2 C_{3} e^{6 t}}{11}, \ z{\left (t \right )} = C_{3} e^{6 t}\right ] \]