87.21.25 problem 25

Internal problem ID [23739]
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 : 25
Date solved : Thursday, October 02, 2025 at 09:44:46 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x+2 y \left (t \right )+2 z \left (t \right )\\ y^{\prime }\left (t \right )&=x+4 y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=-2 x-4 y \left (t \right )-z \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 17

ode:=[diff(x(t),t) = 3*x(t)+2*y(t)+2*z(t), diff(y(t),t) = x(t)+4*y(t)+z(t), diff(z(t),t) = -2*x(t)-4*y(t)-z(t)]; 
ic:=[x(0) = 1, y(0) = 0, z(0) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \\ y \left (t \right ) &= 0 \\ z \left (t \right ) &= -{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 20

ode={D[x[t],t]==3*x[t]+2*y[t]+2*z[t],D[y[t],t]==x[t]+4*y[t]+z[t],D[z[t],t]==-2*x[t]-4*y[t]-z[t]}; 
ic={x[0]==1,y[0]==0,z[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^t\\ y(t)&\to 0\\ z(t)&\to -e^t \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) - 2*y(t) - 2*z(t) + Derivative(x(t), t),0),Eq(-x(t) - 4*y(t) - z(t) + Derivative(y(t), t),0),Eq(2*x(t) + 4*y(t) + z(t) + Derivative(z(t), t),0)] 
ics = {x(0): 1, y(0): 0, z(0): -1} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = e^{t}, \ y{\left (t \right )} = 0, \ z{\left (t \right )} = - e^{t}\right ] \]