79.12.4 problem (d)

Internal problem ID [21113]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter IV. Linear Differential Equations. Excercise XIII at page 189
Problem number : (d)
Date solved : Thursday, October 02, 2025 at 07:08:33 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.070 (sec). Leaf size: 61

ode:=[diff(x(t),t) = -x(t)+y(t)-z(t), diff(y(t),t) = 2*x(t)-y(t)+2*z(t), diff(z(t),t) = 2*x(t)+2*y(t)-z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 149

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

\begin{align*} x(t)&\to \frac {1}{2} e^{-3 t} \left ((2 c_1+c_2-c_3) e^{2 t}-c_2+c_3\right )\\ y(t)&\to \frac {1}{4} e^{-3 t} \left (-2 (2 c_1+c_2-c_3) e^{2 t}+(4 c_1+3 c_2+c_3) e^{4 t}+3 (c_2-c_3)\right )\\ z(t)&\to \frac {1}{4} e^{-3 t} \left (-2 (2 c_1+c_2-c_3) e^{2 t}+(4 c_1+3 c_2+c_3) e^{4 t}-c_2+c_3\right ) \end{align*}

✓ Sympy. Time used: 0.078 (sec). Leaf size: 54

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(-2*x(t) + y(t) - 2*z(t) + Derivative(y(t), t),0),Eq(-2*x(t) - 2*y(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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