79.12.3 problem (c)

Internal problem ID [21112]
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 : (c)
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 )+2 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+y \left (t \right )+2 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: 0.070 (sec). Leaf size: 66

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 134

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

✓ Sympy. Time used: 0.084 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) + y(t) - 2*z(t) + Derivative(x(t), t),0),Eq(x(t) - y(t) - 2*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)
 

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