82.7.1 problem 34-2

Internal problem ID [21857]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-2
Date solved : Thursday, October 02, 2025 at 08:02:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=7 x \left (t \right )-y \left (t \right )+6 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-10 x \left (t \right )+4 y \left (t \right )-12 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=-2 x \left (t \right )+y \left (t \right )-z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 73

ode:=[diff(x(t),t) = 7*x(t)-y(t)+6*z(t), diff(y(t),t) = -10*x(t)+4*y(t)-12*z(t), diff(z(t),t) = -2*x(t)+y(t)-z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 153

ode={D[x[t],t]==7*x[t]-y[t]+6*z[t],D[y[t],t]==-10*x[t]+4*y[t]-12*z[t],D[z[t],t]==-2*x[t]+y[t]-z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{2 t} \left (c_1 \left (-4 e^t+3 e^{3 t}+2\right )-c_2 \left (e^t-1\right )+3 c_3 e^t \left (e^{2 t}-1\right )\right )\\ y(t)&\to -e^{2 t} \left (c_1 \left (-8 e^t+6 e^{3 t}+2\right )+c_2 \left (1-2 e^t\right )+6 c_3 e^t \left (e^{2 t}-1\right )\right )\\ z(t)&\to e^{2 t} \left (-2 c_1 \left (-2 e^t+e^{3 t}+1\right )+c_2 \left (e^t-1\right )+c_3 e^t \left (3-2 e^{2 t}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-7*x(t) + y(t) - 6*z(t) + Derivative(x(t), t),0),Eq(10*x(t) - 4*y(t) + 12*z(t) + Derivative(y(t), t),0),Eq(2*x(t) - y(t) + z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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