87.21.20 problem 20

Internal problem ID [23734]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 178
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:44:42 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=7 x+4 y \left (t \right )-4 z \left (t \right )\\ y^{\prime }\left (t \right )&=4 x-8 y \left (t \right )-z \left (t \right )\\ z^{\prime }\left (t \right )&=-4 x-y \left (t \right )-8 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.067 (sec). Leaf size: 65

ode:=[diff(x(t),t) = 7*x(t)+4*y(t)-4*z(t), diff(y(t),t) = 4*x(t)-8*y(t)-z(t), diff(z(t),t) = -4*x(t)-y(t)-8*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{9 t}+c_3 \,{\mathrm e}^{-9 t} \\ y \left (t \right ) &= \frac {c_2 \,{\mathrm e}^{9 t}}{4}+\frac {c_3 \,{\mathrm e}^{-9 t}}{4}+{\mathrm e}^{-9 t} c_1 \\ z \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{9 t}}{4}+\frac {17 c_3 \,{\mathrm e}^{-9 t}}{4}+{\mathrm e}^{-9 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 133

ode={D[x[t],t]==7*x[t]+4*y[t]-4*z[t],D[y[t],t]==4*x[t]-8*y[t]-z[t],D[z[t],t]==-4*x[t]-y[t]-8*z[t] }; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{9} e^{-9 t} \left (c_1 \left (8 e^{18 t}+1\right )+2 (c_2-c_3) \left (e^{18 t}-1\right )\right )\\ y(t)&\to \frac {1}{18} e^{-9 t} \left (4 c_1 \left (e^{18 t}-1\right )+c_2 \left (e^{18 t}+17\right )-c_3 \left (e^{18 t}-1\right )\right )\\ z(t)&\to \frac {1}{18} e^{-9 t} \left (-4 c_1 \left (e^{18 t}-1\right )-c_2 \left (e^{18 t}-1\right )+c_3 \left (e^{18 t}+17\right )\right ) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-7*x(t) - 4*y(t) + 4*z(t) + Derivative(x(t), t),0),Eq(-4*x(t) + 8*y(t) + z(t) + Derivative(y(t), t),0),Eq(4*x(t) + y(t) + 8*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 4 C_{3} e^{9 t} - \left (\frac {C_{1}}{4} - \frac {C_{2}}{4}\right ) e^{- 9 t}, \ y{\left (t \right )} = C_{1} e^{- 9 t} - C_{3} e^{9 t}, \ z{\left (t \right )} = C_{2} e^{- 9 t} + C_{3} e^{9 t}\right ] \]