52.10.7 problem 7

Internal problem ID [8401]
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 : 7
Date solved : Wednesday, March 05, 2025 at 05:44:40 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= \frac {2 c_3 \,{\mathrm e}^{2 t}}{3}+c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{-t}}{2} \\ y &= c_3 \,{\mathrm e}^{2 t} \\ z \left (t \right ) &= \frac {c_3 \,{\mathrm e}^{2 t}}{3}+c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 88

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

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

✓ Sympy. Time used: 0.107 (sec). Leaf size: 46

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

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