7.23.14 problem 20

Internal problem ID [600]
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 : 20
Date solved : Tuesday, March 04, 2025 at 11:28:29 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 90

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

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

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 164

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

\begin{align*} x(t)\to \frac {1}{9} e^{-t} \left (6 t+3 c_1 \left (e^{3 t}+2\right )+3 c_2 e^{3 t}+3 c_3 e^{3 t}-1-3 c_2-3 c_3\right ) \\ y(t)\to \frac {1}{9} e^{-t} \left (-3 t+3 c_1 \left (e^{3 t}-1\right )+3 c_2 e^{3 t}+3 c_3 e^{3 t}-1+6 c_2-3 c_3\right ) \\ z(t)\to \frac {1}{9} e^{-t} \left (-3 t+3 c_1 \left (e^{3 t}-1\right )+3 c_2 e^{3 t}+3 c_3 e^{3 t}-1-3 c_2+6 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.207 (sec). Leaf size: 75

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

\[ \left [ x{\left (t \right )} = C_{3} e^{2 t} + \frac {2 t e^{- t}}{3} - \left (C_{1} + C_{2} + \frac {1}{9}\right ) e^{- t}, \ y{\left (t \right )} = C_{3} e^{2 t} - \frac {t e^{- t}}{3} + \left (C_{1} - \frac {1}{9}\right ) e^{- t}, \ z{\left (t \right )} = C_{3} e^{2 t} - \frac {t e^{- t}}{3} + \left (C_{2} - \frac {1}{9}\right ) e^{- t}\right ] \]