75.27.7 problem 782

Internal problem ID [17164]
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 20. The method of elimination. Exercises page 212
Problem number : 782
Date solved : Thursday, March 13, 2025 at 09:18:09 AM
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: 0.137 (sec). Leaf size: 63

ode:=[diff(x(t),t) = y(t)+z(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 ) &= 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 &= -2 c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{2 t}-{\mathrm e}^{-t} c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 124

ode={D[x[t],t]==y[t]+z[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}{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: 0.128 (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)
 

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