85.85.1 problem 2

Internal problem ID [23017]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. C Exercises at page 445
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:17:32 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )+z \left (t \right )\\ \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: 0.075 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 124

ode={D[x[t],{t,1}]==y[t]+z[t],D[y[t],{t,1}]==z[t]+x[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}{3} e^{-t} \left (c_1 \left (e^{3 t}+2\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right )\\ y(t)&\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right )\\ z(t)&\to \frac {1}{3} e^{-t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (e^{3 t}+2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 42

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),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)
 

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