69.29.6 problem 807

Internal problem ID [18547]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 807
Date solved : Thursday, October 02, 2025 at 03:14:46 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.155 (sec). Leaf size: 57

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

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

ode={D[x[t],t]==-x[t]+y[t]+z[t],D[y[t],t]==x[t]-y[t]+z[t],D[z[t],t]==x[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}{3} e^{-2 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^{-2 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^{-2 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: 0.075 (sec). Leaf size: 42

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(-x(t) + y(t) - z(t) + Derivative(y(t), t),0),Eq(-x(t) - y(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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