58.19.2 problem 2

Internal problem ID [14924]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.7. Exercises page 375
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:56:36 AM
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 )&=x \left (t \right )+3 y \left (t \right )+z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-3 x \left (t \right )-6 y \left (t \right )+6 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.120 (sec). Leaf size: 73

ode:=[diff(x(t),t) = x(t)-y(t)-z(t), diff(y(t),t) = x(t)+3*y(t)+z(t), diff(z(t),t) = -3*x(t)-6*y(t)+6*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{5 t} \\ y \left (t \right ) &= -\frac {7 c_1 \,{\mathrm e}^{2 t}}{10}-c_2 \,{\mathrm e}^{3 t}-c_3 \,{\mathrm e}^{5 t} \\ z \left (t \right ) &= -\frac {3 c_1 \,{\mathrm e}^{2 t}}{10}-c_2 \,{\mathrm e}^{3 t}-3 c_3 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 217

ode={D[x[t],t]==x[t]-y[t]-z[t],D[y[t],t]==x[t]+3*y[t]+z[t],D[z[t],t]==3*x[t]-6*y[t]+6*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-50 (c_1+c_2)\right )\\ y(t)&\to \frac {1}{45} e^{2 t} \left (5 (c_1+10 c_2) e^{2 t} \cos \left (\sqrt {5} t\right )+\sqrt {5} (7 c_1-11 c_2+9 c_3) e^{2 t} \sin \left (\sqrt {5} t\right )-5 (c_1+c_2)\right )\\ z(t)&\to (c_1+c_2) \left (-e^{2 t}\right )+(c_1+c_2+c_3) e^{4 t} \cos \left (\sqrt {5} t\right )+\frac {(c_1-8 c_2+2 c_3) e^{4 t} \sin \left (\sqrt {5} t\right )}{\sqrt {5}} \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 78

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(-x(t) - 3*y(t) - z(t) + Derivative(y(t), t),0),Eq(3*x(t) + 6*y(t) - 6*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {10 C_{1} e^{2 t}}{3} - C_{2} e^{3 t} - \frac {C_{3} e^{5 t}}{3}, \ y{\left (t \right )} = \frac {7 C_{1} e^{2 t}}{3} + C_{2} e^{3 t} + \frac {C_{3} e^{5 t}}{3}, \ z{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{3 t} + C_{3} e^{5 t}\right ] \]