52.10.27 problem 28

Internal problem ID [8421]
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 : 28
Date solved : Sunday, March 30, 2025 at 01:04:28 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.165 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 0.132 (sec). Leaf size: 54

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

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