87.21.28 problem 28

Internal problem ID [23742]
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 : 28
Date solved : Thursday, October 02, 2025 at 09:44:48 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*}

With initial conditions

\begin{align*} x \left (0\right )&=3 \\ y \left (0\right )&=5 \\ z \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.072 (sec). Leaf size: 40

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)]; 
ic:=[x(0) = 3, y(0) = 5, z(0) = -1]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 46

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={x[0]==3,y[0]==5,z[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 4 e^{9 t}-e^{-9 t}\\ y(t)&\to 4 e^{-9 t}+e^{9 t}\\ z(t)&\to -e^{9 t} \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 37

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 = {x(0): 3, y(0): 5, z(0): -1} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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