69.29.8 problem 809

Internal problem ID [18549]
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 : 809
Date solved : Thursday, October 02, 2025 at 03:14:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 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 )&=y \left (t \right )-2 z \left (t \right )-3 x \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=0 \\ z \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.167 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 2*x(t)-y(t)+z(t), diff(y(t),t) = x(t)+z(t), diff(z(t),t) = y(t)-2*z(t)-3*x(t)]; 
ic:=[x(0) = 0, y(0) = 0, z(0) = 1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode={D[x[t],t]==2*x[t]-y[t]+z[t],D[y[t],t]==x[t]+z[t],D[z[t],t]==y[t]-2*z[t]-3*x[t]}; 
ic={x[0]==0,y[0]==0,z[0]==1}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 1-e^{-t}\\ y(t)&\to 1-e^{-t}\\ z(t)&\to 2 e^{-t}-1 \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) + y(t) - z(t) + Derivative(x(t), t),0),Eq(-x(t) - z(t) + Derivative(y(t), t),0),Eq(3*x(t) - y(t) + 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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