82.6.7 problem 33-40

Internal problem ID [21854]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 33. Systems of ordinary differential equations. Page 1059
Problem number : 33-40
Date solved : Thursday, October 02, 2025 at 08:02:56 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.115 (sec). Leaf size: 70

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 86

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

✓ Sympy. Time used: 0.088 (sec). Leaf size: 76

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

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